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The Mathematics Companion: Mathematical Methods for Physicists and Engineers
1 rating:
3.0
A book by Anthony Craig Fischer-Cripps
This is an interesting and useful little book… .it is very well done, and everything that might be expected to be there is there… .The book might also be invaluable for those undergraduate students in Mathematics, Science or Engineering, … see full wiki
Ideal for the person with a working knowledge of mathematics that needs a refresher
The prime audience for this book is working professionals whose education required at least the first two years of the math major. To be more specific, I mean people that had to take at least a two-semester college sequence of calculus, linear algebra and a few other courses and now use that knowledge in their jobs. The author is a physicist, so the topics reviewed are those that a working physicist would need to have studied, although people in other areas such as engineering and chemistry will also find it valuable. A wide variety of topics are briefly surveyed:
There is almost nothing in the way of theory, the topic is presented and a few examples/applications are demonstrated. It is assumed that the reader has been exposed to the material in detail and needs only a reminder or a slightly more in-depth refresher. While there are some typos, there really are no more than in most books. The problem is that mathematics is presented in such condensed form; most errors of a single character are magnified. Nevertheless, this is a book that the person working in a profession based on mathematics can use as a valuable aid when a quick update is needed.
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Syllabus in Solid Geometry
This course is a study of plane and solid geometry it includes precise definitions, high school course syllabus october 2013 2 lifelong learning standards
Geometrical and mechanical engineering drawing syllabus class discussions and illustrate the use and purpose of solid geometry in mechanical engineering
Basic geometry course syllabus cindy kroon course description the main focus of geometry is on plane and solid figures and their properties a major emphasis is
Syllabus for math 1325 analytic geometry this course satisfies the university of texas at arlington core curriculum requirement in mathematics
Geometry honors course syllabus middleboro high school, ms miles of a 3 d solid, and solve problems involving a solid, given its net students will solve
Integrated geometry course includes an in depth analysis of plane, solid, and coordinate geometry as they integrated geometry a course syllabus
Geometry honors syllabus course description the geometry course includes an in‐depth analysis of plane, solid, and coordinate geometry as they relate to
A solid understanding of linear algebra and a liking for abstract mathematics syllabus mathematics 106 treats difierential geometry of curves in the plane
Honors geometry syllabus 2013 14 formulas pertaining to the measurement of plane and solid figures coordinate geometry and two and three dimensional geometry
Geometry b www studying the volume and surface area of solid objects in this unit, there will be: geometry b syllabus rev final071712docx author:
34 solve specific geometrical problems in plane and solid geometry five questions will be set on applied geometry taken from the syllabus block plans, site
Syllabus part i: plane and solid geometry scales construction and use of plain and diagonal scales: selection and use of suitable scales construction and use of
Math 96 syllabus page 1 of 5 intermediate algebra and geometry is the second of a two semester integrated sequence in algebra and solid geometry
Performance of malaysian secondary school students in geometry was still worrying in the trends in international mathematics and science
Modern abc of physics, class xii solid geometry : syllabus for practical exams will be same as topics covered in theory for that term
Syllabus for topics in geometry summer 2002 as to geometry, discrete geometry, and solid modeling, the main focus of
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Math, students wil.Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 176 pages. 0.272. Bookseller Inventory # 978,.Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability. 176 pages. 0.272. Bookseller Inventory # 97800728858
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Michael Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Serieshas evolved to meet today's course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors.
Product Details
ISBN-13: 9780321832139
Publisher: Pearson
Publication date: 3/7/2012
Edition number: 6
Pages: 1152
Sales rank: 72,593
Product dimensions: 8.60 (w) x 11.10 (h) x 0.70 (d)
Meet the Author
Michael Sullivan, Emeritus Professor of Mathematics at Chicago State University, received a Ph.D. in mathematics from the Illinois Institute of Technology. Mike taught at Chicago State for 35 years before recently retiring. He is a native of Chicago's South Side and divides his time between a home in Oak Lawn IL and a condo in Naples FL.
Mike is a member of the American Mathematical Society and the Mathematical Association of America. He is a past president of the Text and Academic Authors Association and is currently Treasurer of its Foundation. He is a member of the TAA Council of Fellows and was awarded the TAA Mike Keedy award in 1997 and the Lifetime Achievement Award in 2007. In addition, he represents TAA on the Authors Coalition of America.
Mike has been writing textbooks for more than 35 years and currently has 15 books in print, twelve with Pearson Education. When not writing, he enjoys tennis, golf, gardening, and travel.
Mike has four children: Kathleen, who teaches college mathematics; Michael III, who also teaches college mathematics, and who is his coauthor on two precalculus series; Dan, who is a sales director for Pearson Education; and Colleen, who teaches middle-school and secondary school mathematics. Twelve grandchildren round out the family.
Mike Sullivan III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf
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Computer graphics is important in many areas including engineering design, architecture, education, and computer art and animation. This book examines a wide array of current methods used in creating real-looking objects in the computer, one of the main aims of computer graphics. Key features: - Good foundational mathematical introduction to curves and surfaces; no advanced math required - Topics organized by different interpolation/approximation techniques, each technique providing useful information about curves and surfaces - Exposition motivated by numerous examples and exercises sprinkled throughout, aiding the reader - Includes a gallery of color images, Mathematica code listings, and sections on curves and surfaces by refinement and on sweep surfaces - Web site maintained and updated by the author, providing readers with errata and auxiliary material This engaging text is geared to a broad and general readership of computer science/architecture engineers using computer graphics to design objects, programmers for computer gamemakers, applied mathematicians, and students majoring in computer graphics and its applications. It may be used in a classroom setting or as a general reference.
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Read an Excerpt
Author Information
Mark Ryan, a graduate of Brown University and the University of Wisconsin Law School, has been teaching math since 1989. He runs the Math Center in Winnetka, Illinois ( where he teaches high school math courses including an introduction to geometry and a workshop for parents based on a program he developed, The 10 Habits of Highly Successful Math Students. In high school, he twice scored a perfect 800 on the math portion of the SAT, and he not only knows mathematics, he has a gift for explaining it in plain English. He practiced law for four years before deciding he should do something he enjoys and use his natural talent for mathematics. Ryan is a member of the Authors Guild and the National Council of Teachers of Mathematics. Geometry Workbook For Dummies is Ryan's fourth book. Everyday Math for Everyday Life was published in 2002, Calculus For Dummies (Wiley) in 2003, and Calculus Workbook For Dummies in 2005. His math books have sold over 100,000 copies.
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Overview
Main description
A no-nonsense, practical guide to help you improve your algebra II skills with solid instruction and plenty of practice, practice, practice
Practice Makes Perfect: Algebra II presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations, as well as an introducing you to probability and trigonometry. Inside you will find the help you need for boosting your skills, preparing for an exam or re-introducing yourself to the subject. More than 500 exercises and answers covering all aspects of algebra will get you on your way to mastering algebra!
Author comments
Christopher Monahan is a retired teacher of math with more than 30 years of classroom experience at the college and high school level. As a member of the Association of Mathematics Teachers of New York State, Monahan served as president (2009-2010) and vice president (2006-2007). He served on a number of committees for the New York State Department of Education for writing Regents Exam questions for Math A and Math B, for determining scope and range for the Geometry Regents. He has been a National Trainer for Texas Instruments in Teaching Teachers with Technology since 2001.
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Prentice Hall Math Course 1 Student Workbook 2007c
Prentice Hall Mathematics Course 1: A combination of rational numbers, patterns, geometry and integers in preparation for one- and two-step equations ...Show synopsisPrentice Hall Mathematics Course 1: A combination of rational numbers, patterns, geometry and integers in preparation for one- and two-step equations and inequalities.Guided Problem Solving strategies throughout the text provide students with the tools they need to be effective and independent learners. An emphasis on fractions solidifies student understanding of rational number operations preparing them to apply these skills to algebraic equations. Activity Labs throughout the text provide hands-on, minds-on experiences reaching all types of learners
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More About
This Textbook
Overview
Despite the increasing use of computers, the basic need for mathematical tables continues. Tables serve a vital role in preliminary surveys of problems before programming for machine operation, and they are indispensable to thousands of engineers and scientists without access to machines. Because of automatic computers, however, and because of recent scientific advances, a greater variety of functions and a higher accuracy of tabulation than have been available until now are required.
In 1954, a conference on mathematical tables, sponsored by M.I.T. and the National Science Foundation, met to discuss a modernization and extension of Jahnke and Emde's classical tables of functions. This volume, published 10 years later by the U.S. Department of Commerce, is the result. Designed to include a maximum of information and to meet the needs of scientists in all fields, it is a monumental piece of work, a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems.
The book contains 29 sets of tables, some to as high as 20 places: mathematical constants; physical constants and conversion factors (6 tables); exponential integral and related functions (7); error function and Fresnel integrals (12); Bessel functions of integer (12) and fractional (13) order; integrals of Bessel functions (2); Struve and related functions (2); confluent hypergeometric functions (2); Coulomb wave functions (2); hypergeometric functions; Jacobian elliptic and theta functions (2); elliptic integrals {9); Weierstrass elliptic and related functions; parabolic cylinder functions {3); Mathieu functions (2); spheroidal wave functions (5); orthogonal polynomials (13); combinatorial analysis (9); numerical interpolation, differentiation and integration (11); probability functions (ll); scales of notation {6); miscellaneous functions {9); Laplace transforms (2); and others.
Each of these sections is prefaced by a list of related formulas and graphs: differential equations, series expansions, special functions, and other basic relations. These constitute an unusually valuable reference work in themselves. The prefatory material also includes an explanation of the numerical methods involved in using the tables that follow and a bibliography. Numerical examples illustrate the use of each table and explain the computation of function values which lie outside its range, while the editors' introduction describes higher-order interpolation procedures. Well over 100 figures illustrate the text.
In all, this is one of the most ambitious and useful books of its type ever published, an essential aid in all scientific and engineering research, problem solving, experimentation and field work. This low-cost edition contains every page of the original government publication. Preface by A. V. Astin. Foreword by Advisory Committee, Conference on Mathematical Tables. Editors' Introduction. Indices to Subjects, Not
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Edgenuity Adds Algebra I for Blended Classrooms
"The full-year course focuses on the key areas of the Common Core model pathway for Algebra I and builds on the foundation established in middle grades by deepening students' understanding of linear and exponential functions, and developing fluency in writing and solving one-variable equations and inequalities," according to a news release.
Features include:
On-screen teachers thinking aloud to model problem solving;
Guiding questions for each lesson to promote inquiry and a focus on bigger ideas;
Performance tasks, designed to allow students to show understanding with real-world applications.
"Algebra is critical to student success in college and career because it sharpens essential reasoning skills," said Sari Factor, CEO of Edgenuity, in a prepared statement. "Our new Algebra I course raises the bar for accessibility, using multimedia and interactives to develop analytical problem-solving skills and to ensure mastery for students with a variety of learning styles
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Algebra IIUnit 2Polynomial Equations and Inequalities
Activity 2: Using Pascal's Triangle to Expand Binomials (GLEs: Grade 10: 24, 26; Grade
11/12: 2, 8, 27)
Materials List: paper, pencil, graphing calculator, 5 transparencies or 5 large sheets of paper,
Expanding Binomials Discovery Worksheet BLM, Expanding Binomials Discovery Worksheet
BLM Key
The focus of this activity is to find a pattern in coefficients in order to quickly expand a binomial
using Pascal's triangle, and to use the calculator nCr button to generate Pascal's triangle.
Math Log Bellringer: Expand the following binomials:
(1) (a + b)0
(2) (a + b)1
(3) (a + b)2
(4) (a + b)3
(5) (a + b)4
(6) Describe the process you used to expand #5
Solutions:
(1) 1, (2) a + b,(3) a2 + 2ab + b2,(4) a3 + 3a2b + 3ab2 + b3,
(5)a4 + 4a3b + 6a2b2+4ab3 + b4, (6) Answers will vary.
Activity:
Have five of the students each work one of the Bellringer problems on a transparency or
large sheets of paper, while the rest of the students work in their notebooks. Have the five
students put their answers in front of the class and explain the process they each used.
Compare answers to check for understanding of the FOIL process.
Write the coefficients of each Bellringer problem in triangular form (Pascal's triangle) and
have students find a pattern.
Expanding Binomials Discovery Worksheet:
On this worksheet, the students will discover how to expand a binomial using both Pascal's
triangle and combinations. Distribute the Expanding Binomials Discovery Worksheet BLM
and have students work in pairs on the Expanding Binomials section of the worksheet.
Circulate to check for understanding and stop after this section to check for correctness.
Allow students to complete the section on Using Combinations to Expand Binomials and
check for correctness.
Administer the ActivitySpecific Assessment to check for understanding expanding a binomial.
Algebra IIUnit 2Polynomial Equations and Inequalities
Activity-Specific Assessments
Activity 2:
Draw Pascal's triangle to the row containing 5, then expand the following binomials: (1)
(x y)5
(2) (4x + y)3
Solutions:
(1) x5 5x4y + 10x3y2 10x2y3 + 5xy4 y5
(2) 64x3 + 48x2y + 12xy2 + y
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Using discrete dynamical systems, this book introduces powerful mathematical modeling techniques, both standard analytical and modern computational, to students in mathematics, the natural sciences, and the social sciences. With minimal mathematical background, students will quickly progress from the traditional study of exponential growth and decay that simple linear equations always exhibit, to an investigation of recently discovered chaotic dynamics often associated with nonlinear systems. A wide diversity of applications demonstrates the usefulness and relevance of topics that have often been viewed as excessively theoretical or abstract, such as sequences, limits, linear algebra, complex variables, and more. By taking advantage of discrete dynamical systems, students will have the opportunity to experience some fascinating areas of mathematical discovery.
Features
Introduces exciting areas of current research, such as dynamical systems, that show that mathematics is a vibrant and evolving discipline.
Emphasizes the determination of the dynamics of solutions as the primary focus beginning in Chapter 3, since this is intended for a course in discrete dynamical systems and not differential equations.
Begins Chapters 2 through 5 with the construction of a host of simple iterative models of the type that is to be investigated later in that chapter. These models motivate the analysis that is to follow and ensures that applications will not be sacrificed if time runs short.
Contains an ample supply of exercises at varying levels of difficulty with many answers provided, and numerous suggested computer projects with specific instructions for their completion.
Includes an appendix that explains how most computer projects can be done using either a spreadsheet program such as Microsoft Excel? or the powerful software package of Mathematica?.
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Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
9780471476023
ISBN:
0471476021
Pub Date: 2005 Publisher: John Wiley & Sons Inc
Summary: Did you know that games and puzzles have given birth to many of today's deepest mathematical subjects? Now, with Douglas Ensley and Winston Crawley's Introduction to Discrete Mathematics, you can explore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems. You will discover how new mathematical topics can ...be applied to everyday situations, learn how to work with proofs, and develop your problem-solving skills along the way. Online applications help improve your mathematical reasoning. Highly intriguing, interactive Flash-based applications illustrate key mathematical concepts and help you develop your ability to reason mathematically, solve problems, and work with proofs. Explore More icons in the text direct you to online activities at Improve your grade with the Student Solutions Manual. A supplementary Student Solutions Manual contains more detailed solutions to selected exercises in the text.
Ensley, Douglas E. is the author of Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games, published 2005 under ISBN 9780471476023 and 0471476021. Two hundred twenty six Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games textbooks are available for sale on ValoreBooks.com, sixty six used from the cheapest price of $125.80, or buy new starting at $155.05.[read more]
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). Almost new condition. SKU:9780471476023-2-0-3 Orders ship the same or next business day. Expedite... [more]RENTAL: Supplemental materials are not guaranteed (access codes, DVDs, workbooks). Almost80471476023-2-0-3 Orders ship the same or next business day. Expedite... [more]1476023
ISBN:0471476021
Pub Date:2005 Publisher:John Wiley & Sons Inc
Valore Books is the best place for cheap Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games rentals, or used and new condition books that can be mailed to you in no time.
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MA 441 - Intro to Mathematical Analysis
MA 441 - Intro to Mathematical Analysis
This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Topics include fundamental mathematical logic and set theory, the real number systems, sequences, limits and completeness, elements of topology, continuity, derivatives and related theorems, Taylor expansions, the Riemann integral, and the Fundamental Theorem of Calculus.
Prerequisites: MA 227
Course Objectives
MA441 and MA442 are capstone courses for mathematics majors. The two courses supply basic preparation for either graduate school or for application of mathematics to problems of science and engineering.
Learning Outcomes
There are a number of different approaches to this course. The exact content of the course is not as important as the development of mathematical maturity by the students. Accordingly course outcomes are based on Bloom's taxonomy.
Know: Recall definitions and statements of theorems.
Comprehend: Be able explain and restate theorems and definitions in different contexts and as they apply to special cases.
Apply: Use the theorems and techniques taught in the course to solve problems.
Analyze: Recognize which theorems and definitions apply to various situations.
Synthesize: Be able to construct proofs.
The outcomes are repeated for each of the three major divisions of the course, namely
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Math : Facing an American Phobia (98 Edition)
by Marilyn Burns Publisher Comments
Challenging negative attitudes and delivering a positive message about what math can and should mean to all of us, this resource is both an entertaining and invaluable read. From "Talking Turkey About Arithmetic" to "Making Math Make... (read more)
Calculus: Early Transcendentals
by James Stewart Publisher Comments
Success in your calculus course starts here!... (read more)
Fundamentals of Differential Equations (8TH 12 Edition)
by R. Kent Nagle Publisher Comments
Fundamentals of Differential Equations presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. Available in two versions, these flexible texts offer the instructor many choices in... (read more)
Complexity: The Emerging Science at the Edge of Order and Chaos
by M Mitchell Waldrop Publisher Comments... (read more)
Algorithms (08 Edition)
by Sanjoy Dasgupta Publisher Comments
Papadimitrio is a one term algorithms text that takes an integrated approach and is priced well below any of the competitors within the market at $30.00.This product will take a more frequent revision cycle and will be the most current and up-to-date
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Intermediate Algebra: Connecting Concepts through Applications
9780534496364
ISBN:
0534496369
Edition: 1 Pub Date: 2011 Publisher: Brooks Cole
Summary: INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate so...lutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and skills.
Clark, Mark is the author of Intermediate Algebra: Connecting Concepts through Applications, published 2011 under ISBN 9780534496364 and 0534496369. Five hundred seventy four Intermediate Algebra: Connecting Concepts through Applications textbooks are available for sale on ValoreBooks.com, two hundred twenty one used from the cheapest price of $42.99, or buy new starting at $122.69
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The Third Edition of the Bittinger Graphs and Models series helps students succeed in algebra by emphasizing a visual understanding of concepts. This latest edition incorporates a new Visualizing the Graph feature that helps students make intuitive connections between graphs and functions without the aid of a graphing calculator.
In addition, students learn problem-solving skills from the Bittinger hallmark five-step problem-solving process coupled with Connecting the Concepts and Aha! Exercises.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
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Overview
Overview
Exercises that require students to engage in a comparison reasoning process: students rank variations of a physical situation on the basis of a specified physical quantity and explain their reasoning. Ranking Tasks frequently elicit students' natural ideas about the behavior of physical systems rather than a memorized response, providing instructors with a way to gain important insights into students' thinking.
Type of Method
Curriculum supplement
Level
Designed for:
Intro College Calculus-based, Intro College Algebra-based
Can be adapted for: Teacher Preparation, Teacher Professional Development, High School, Intro College Conceptual
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Curve Tracing in Calculus
Topic page: multivariable calculus traces, level curves, and contuour maps click here for a printable pdf version traces the trace of a surface in a plane is the
Vector calculus i year btech 33 curve tracing cartesian co ordinates is position vector and is the curve
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Conic sections and curve tracing calculus, differential and integral, is introduced with an intuitive approach and with emphasis on real life applications,
The objective of calculus is for students to learn the basics of the calculus concavity and convexity of a curve, points of inflection, curve tracing:
Differential calculus applications curve tracing: (vertical) extent of the curve is determined by the intervals of x(y) for which the curve exists
Ap calculus bc course design and philosophy students enrolled in calculus bc have completed calculus ap ab or an accelerated math course during their sophomore or
27 calculus —– newton and leibniz in his tract, newton thought of a particle tracing out a curve with two moving lines which were the coordinates
Finding limits of a piecewise defined function calculus i tutorial, by dave collins i from the graph ii from the algebraic representation of the function
Ap calculus ab course design and it "reappears" when the tracing continues at x = 21 students can slope curve follows the path of the cosine function
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Differential calculus, in fine type, as a footnote to the chapter on envelopes and a long one on curve tracing and on properties of special curves
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Personal technology and the calculus graphical explorations are not limited to curve tracing, however the casio cfx 9850 g includes a number of
The derivative and its application to curve tracing and max min theory antiderivative, area under a curve student solutions manual for tan's applied calculus
Minima of functions of two variable, curve tracing 16 2 integral calculus integration as a limit of a sum, rectification, surface area of revolution,
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Curve tracing general rules to trace cartesian, • analyze position, velocity, and acceleration in two or three dimensions using the calculus
Program : b sc in mathematics three years course : calculus i year / b sc general (mpc)
Graphing trigonometric functions using a ti 83+ – mr ruby page 6 now, we can graph the y=sec(x) function and we should see: try tracing this function to values of
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44 calculus 4 plot the curve a^ 4 y* = 81, taking 1 cm as the unit curve tracing in the early work of plotting curves from their equations the only way we had
Using temath's visualization tools in calculus 6 figure 8 drawing tangents to the sine curve as we draw these tangents, their slopes are computed and written into the
Calculus for the life sciences ii area under a curve 3 riemann integral quite accurately by a simple scanning or tracing process
Mathematics for computer graphics ray tracing iii dr philippe b laval kennesaw state university november 12, 2003 abstract this document is a continuation of the
Parametric and polar equations with a figure skater note: you may notice differences between this maple worksheet and the equivalent mathematica notebook
| 677.169 | 1 |
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Learn and practice your Algebra II skills with two friendly For Dummies guides for one low price!This value-priced, two-book bundle brings two popular For Dummiesmath guides together to offer readers essential Algebra Iinstruction combined with ...
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Most people agree that math is important, but few would say it's fun. This book will show you that the subject you learned to hate in high school can be as entertaining as a witty remark, as engrossing as the mystery novel you can't put down--in ...
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A user-friendly, hands-on guide to recognizing and conducting proper research techniques in data collection Offering a unique approach to numerical research methods, Analyzing Quantitative Data: An Introduction for Social Researchers presents ...
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Learn and practice yourBasic Math & Pre-Algebraskills with two friendly For Dummies guides for one low price!This value-priced, two-book bundle brings two popular For Dummiesmath guides together to offer readers essentialBasic Math & ...
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Just about everyone takes a geometry class at one time or another. And while some people quickly grasp the concepts, most find geometry challenging. Covering everything one would expect to encounter in a high school or college course, Idiot's ...
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Straightforward Statistics: Understanding the Tools of Research is a clear and direct introduction to statistics for the social, behavioral, and life sciences. Based on the author's extensive experience teaching undergraduate statistics, this ...
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Learn and practice your Algebra I skills with two friendly For Dummies guides for one low price!This value-priced, two-book bundle brings two popular For Dummiesmath guides together to offer readers essential Algebra Iinstruction combined with ...
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An extensive collection of videos demonstrating all aspects of using Camtasia to capture screen videos to producing...
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An extensive collection of videos demonstrating all aspects of using Camtasia to capture screen videos to producing them. Overview the basic screen/video capture process is presented. Information of where the software can be obtained online is given. The user will also explore the different ways you can capture graphics, text, screens, and web site graphics using this product.
"This virtual manipulative allows you to solve simple linear equations through the use of a balance beam. Unit blocks...
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"This virtual manipulative allows you to solve simple linear equations through the use of a balance beam. Unit blocks (representing 1s) and X-boxes (for the unknown, X), are placed on the pans of a balance beam. Once the beam balances to represent the given linear equation, you can choose to perform any arithmetic operation, as long as you DO THE SAME THING TO BOTH SIDES, thus keeping the beam balanced. The goal, of course, is to get a single X-box on one side, with however many unit blocks needed for balance, thus giving the value of X."
This site "provides an eclectic mix of sound, science, and Incan history in order to raise students' interest in Euclidean...
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This site "provides an eclectic mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, 'scientific speculation', and more."
This site is a comprehensive introduction to tessellations. The basic mathematics underlying tessellations are explained and...
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This site is a comprehensive introduction to tessellations. The basic mathematics underlying tessellations are explained and many examples of tessellations in real life are displayed. M. C. Escher and his well-known tessellations are also covered and animated.
A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video...
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A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction.
This site walks you through the creation of your first Camtasia video, converting AVI format videos to Flash format for...
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This site walks you through the creation of your first Camtasia video, converting AVI format videos to Flash format for streaming in the Internet Explorer Browser, and how to publish the resulting Flash video on Blackboard or a website.
demonstrates real-world applications of math to all those students who say, "How will I ever use this?" You don't get much...
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demonstrates real-world applications of math to all those students who say, "How will I ever use this?" You don't get much more real world than solving the problems of dividing estates fairly, apportioning legislative seats, or cutting a cake in even pieces. Each activity includes printable worksheet materials as you incorporate this standards-based subject--discrete mathematics--in your math classes (grade 9 and up).
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97803215679706Mathematics All Around
Mathematics All Around
Mathematics All Around
Mathematics All Around
Mathematics All Around Plus MyMathLab Student Access Kit
Mathematics All Around Plus NEW MyMathLab with Pearson eText -- Access Card Package
Student Solutions Manual for Mathematics All Around
Videos on DVD with Optional Subtitles for Mathematics All Around
Summary
Normal 0 false false false MicrosoftInternetExplorer4 Mathematics All Around, Fourth Edition, is the textbook for todayrs"s liberal arts mathematics students. Tom Pirnot presents math in a way that is accessible, interesting, and relevant. Like having a teacher on call, its clear, conversational writing style is enjoyable to read and focuses on helping students understand the math, not just get the correct answers on the test. Useful features throughout the book enable students to become comfortable with thinking about numbers and interpreting the numerical world around them. Problem Solving: Strategies and Principles; Set Theory: Using Mathematics to Classify Objects; Logic: The Study of What's True or False or Somewhere in Between; Graph Theory (Networks): The Mathematics of Relationships; Numeration Systems: Does It Matter How We Name Numbers?; Number Theory and the Real Number System: Understanding the Numbers All Around Us; Algebraic Models: How Do We Approximate Reality?; Modeling with Systems of Linear Equations and Inequalities: What's the Best Way to Do It?; Consumer Mathematics: The Mathematics of Everyday Life; Geometry: Ancient and Modern Mathematics Embrace; Apportionment: How Do We Measure Fairness?; Voting: Using Mathematics to Make Choices; Counting: Just How Many Are There?; Probability: What Are the Chances?; Descriptive Statistics: What a Data Set Tells Us For all readers interested in mathematics.
Author Biography
Tom Pirnot received his bachelor's degree in music from Wilkes College and his PhD in mathematics from The Pennsylvania State University. He taught both mathematics and computer science at Kutztown University for thirty eight years. He has long been an innovator in liberal arts mathematics, writing his first text Mathematics: Tools and Models with Dalton Hunkins in 1977 which introduced topics such as apportionment, graph theory, and modeling to liberal arts students. His current text, Mathematics All Around, is now in its fourth edition. Tom continues to enjoy the loving support and encouragement of his wife Ann, their four children, and three grandchildren.
Table of Contents
Problem Solving: Strategies and Principles
Problem Solving
Inductive and Deductive Reasoning
Estimation
Set Theory: Using Mathematics to Classify Objects
The Language of Sets
Comparing Sets
Set Operations
Survey Problems
Looking Deeper: Infinite Sets
Logic: The Study of What's True or False or Somewhere in Between
Statements, Connectives, and Quantifiers
Truth Tables
The Conditional and Biconditional
Verifying Arguments
Using Euler Diagrams to Verify Syllogisms
Looking Deeper: Fuzzy Logic
Graph Theory (Networks): The Mathematics of Relationships
Graphs, Puzzles, and Map Coloring
The Traveling Salesperson Problem
Directed Graphs
Looking Deeper: Scheduling Projects Using PERT
Numeration Systems: Does It Matter How We Name Numbers?
The Evolution of Numeration Systems
Place Value Systems
Calculating in Other Bases
Looking Deeper: Modular Systems
Number Theory and the Real Number System: Understanding the Numbers All Around Us
Number Theory
The Integers
The Rational Numbers
The Real Number System
Exponents and Scientific Notation
Looking Deeper: Sequences
Algebraic Models: How Do We Approximate Reality?
Linear Equations
Modeling with Linear Equations
Modeling with Quadratic Equations
Exponential Equations and Growth
Proportions and Variation
Functions
Looking Deeper: Dynamical Systems
Modeling with Systems of Linear Equations and Inequalities: What's the Best Way to Do It?
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Mathematics 4 Introduction Purpose of this Document This document is being made available to Idaho School Districts for the purpose of assisting in the preparation of students for the Grade10 Mathematics ISAT.
In the following formulas, n represents the number of sides. • In a polygon, the sum of the measures of the interior angles is equal to 180 ... MATHEMATICS GRADE SUNSHINE STATE STANDARDS TEST BOOK Released: August 2006 10 Last used: Summer 2006
Grade10 students taking the 2011 FCAT Mathematics test and all Retake students taking the ... In the following formulas, n represents the number of sides. • In a polygon, the sum of the measures of the interior angles is equal to
6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); ... Mathematics released test questions for the California Standards Test, grade six. Created Date: 1/11/2008 10:50:09 AM ...
SECTION 3 - SUMMARY OF KEY WATER PLANT OPERATOR MATH FORMULAS Most mathematics problems found on Grade I - IV level examinations are process control problems that are responsive to a host of math formulas that have been developed over
Grade 6 Includes: • Georgia Performance Standards for Grade 6 • Diagnostic Test • Practice for Each Grade 6 ... • Become familiar with a variety of formulas and when they should be used. • Make sure that the number of the question on the answer sheet
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Discovering Geometry Supplements
The Discovering Geometry textbook package includes all of the teaching resources necessary to enrich learning and to make the teaching experience more rewarding. In addition to a comprehensive, wraparound-style Teacher's Edition, you'll receive a spectrum of evaluation and assessment tools that let all students demonstrate what they have learned. Broad input from teachers has made the teaching resources package a valuable asset for teachers implementing this exciting curriculum.You can purchase Discovering Geometry textbook publications through our publisher Kendall Hunt.
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This book is devoted to the lectures presented at the Sixth International Conference on Formal Power Series and Algebraic Combinatorics held at DIMACS in May 1994. The conference attracted approximately 180 graduate students and junior and senior researchers from all over the world.
Generally speaking, algebraic combinatorics involves the use of techniques from algebra, algebraic topology, and algebraic geometry in solving combinatorial problems; or it involves using combinatorial methods to attack problems in these areas. Combinatorial problems amenable to algebraic methods can arise in these or other areas of mathematics, or in areas such as computer science, operations research, physics, chemistry, and, more recently, biology.
Because of this interplay among many fields of mathematics and science, algebraic combinatorics is an area in which a wide variety of ideas and methods come together. The papers in this volume reflect the interesting aspects of this rich interaction.
Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with the Association for Computer Machinery (ACM).
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An overview of developments in the past 15 years of adjunction theory, the study of the interplay between the intrinsic geometry of a projective variety and the geometry connected with some embedding of the variety into a projective space. Topics include consequences of positivity, the Hilbert schem more...
This book presents a complete and accurate study of algebraic circuits, digital circuits whose performance can be associated with any algebraic structure. The authors distinguish between basic algebraic circuits, such as Linear Feedback Shift Registers (LFSRs) and?cellular automata and algebraic circuits, such as finite fields or Galois fields.The... more...
The volume consists of invited refereed research papers. The contributions cover a wide spectrum in algebraic geometry, from motives theory to numerical algebraic geometry and are mainly focused on higher dimensional varieties and Minimal Model Program and surfaces of general type. A part of the articles grew out a Conference in memory of Paolo Francia... more...
This new-in-paperback edition provides an introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. Clear explanations of both theory and applications, and almost 600 exercises are included in the text. - ;This new-in-paperback... more...
Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck's schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly... more...
Algebraic Geometry provides an impressive theory targeting the understanding of geometric objects defined algebraically. Geometric Modeling uses every day, in order to solve practical and difficult problems, digital shapes based on algebraic models. This book is a collection of articles bridging these two areas. more...
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People often think that "algebra" sounds like a big and scary thing, but I think that's mostly just because they don't understand what it actually is. Which is a real shame because algebra is incredibly useful in the real world. Plus, it turns out that the big ideas behind it all are actually fairly straight-forward. So today we're going to combat this universal fear and misunderstanding of an otherwise beautiful subject by diving in and answering the big question: What is algebra?
Sponsor: This episode is brought to you by GoDaddy.com. GoDaddy.com offers everything you need to make a name for yourself on the Web, including domain name registration, website hosting, and more. Get 20% off your hosting plan purchase at GoDaddy.com by using the code hostpod60 at checkout.
The Problem with Algebra
The problem with algebra is that it isn't a simple tangible object…like a spoon. If it was, it'd be easy to explain. After all, few people who manage to eat lunch with a spoon will have trouble explaining how it works afterwards. But, unfortunately, that's just not what algebra is. So, what is it then? Well, instead of being a thing like a spoon, algebra is more analogous to being something like a style or a method of using a spoon...
I'm thinking of a sensible two-fingered (plus a thumb) grip versus a crazy five-fingered silverware stranglehold. Sure, both styles will get the job done, but one technique is definitely more elegant and efficient than the other.
The same is true with "style" in math. One type of method—sometimes algebraic—is usually most appropriate for solving a particular type of problem. But styles are much harder to recognize, point at, and label than things—which is why recognizing and labeling algebra can be tricky. So then what's the best way to think of algebra?
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A compendium of mathematical definitions, formulas, figures, tabulations, and references. Provides a highly readable, distinctive text applicable to a broad spectrum of readers with a diverse range of mathematical backgrounds and interests. Draws connections to other areas of mathematics and science and uses Mathematica examples to illustrate concepts.
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{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":125.55,"ASIN":"0130144126","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":119.85,"ASIN":"0471433314","isPreorder":0}],"shippingId":"0130144126::fgDv3Kgs09xtL2Ed2G8B%2Be1MNY%2FG0GN1d8aFrpz3dMjARDhWxzm7wswOw2d%2Fp108mmen8J5ugx7CgrI6cPQaSkWY0L%2F1XITPTirnpmPPIKk%3D,0471433314::ntjHWAKuZXHOD3AxzLYOnulaEwptKbA46nXUSrtTwEYvPJKn%2BjCVIWiRSd6px02CICTuSLaU2tftnV8uaFldEJHI99a%2FCrTR%2BGc1W20rUG45gLdsUu5TXOffering a survey of both discrete and continuous mathematics, Mathematical Thinking begins with the fundamentals of mathematical language and proof techniques such as induction. These are applied to easily-understood questions in elementary number theory and counting. Further techniques of proofs are then developed via fundamental topics in discrete and continuous mathematics. The text can be used for courses emphasizing discrete mathematics, continuous mathematics, or a balance between the two. It contains many engaging examples and stimulating exercises.
--This text refers to an out of print or unavailable edition of this title.
From the Back CoverMost Helpful Customer Reviews
This book was an excellent read, and provided some great information. However, as a math text, I feel like it should have provided a bit more examples, and perhaps even a solutions manual.
I understand that the books main focus was on the abstract discussions of mathematics, but I feel like that should merit the writer to put a bit more examples to drive home the theorems before copious amounts of problems are assigned at the end of each chapter. A ton of these problems are classical, and need to be understood, in light of this, why isn't there a solutions manual to better explain these problems?
I greatly appreciated the voices of the writers keeping themselves grounded in pragmatic language. Too often will mathematicians get lofty in their dictions and fuddle the material they claim to understand all too well. This book did a great job discussing simple concepts simply, meanwhile working the more difficult ones with more space.
I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores. I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar "proofs and problem solving" style books.
I ran into the first edition of this book ten years ago when taking courses at George Mason University, and really loved it. I still love it. It covers proofs from all basic "pieces" of mathematics and gives the reader a good feel for the "proofology," both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and abstract algebra courses. It's not perfect though. I haven't bought the second edition, but in the first edition, Example 2.21, p.27 says: "An integer is even if and only if it is the sum of two odd integers." Obviously, it is easy to show the sum of two odd integers is even by forming the sum (2k+1) + (2l+1) = 2(k+l+1), which is twice an integer and thus even. But, if an integer is even, it can be the sum of two odd integers OR two even integers, so the statement is not complete. If small stuff like that doesn't bother you, this book is for you.
The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.
My main problem with this text is the level of understanding it assumes on behalf of the reader. Many reviewers say "great reference" or something of the sort. But this book is a *foundational* text. It's not a book for mathematicians or a mathematically mature reader -- they should already own the techniques presented here. A book of this kind should be suitable for self study and this book fails in that department. Given the amount of math it assumes, I would imagine those at that level are already fairly assimilated to proofs and the like. Hence my critique that the level and function is confused. Note also that this book is *expensive*.
If you want a book on problem solving, go with Zeitz or Engel, or something of "olympiad" character. If you want a book to learn proof techniques, Vellman or Eccles is good; Solow for true beginners.
I originally purchased this book as a text for a math course and quite enjoyed the selections that we worked through in it. It has been about a year since I took that course and I still find myself going back for references in the book. It is a must have for someone who is interested in proofs or will be doing them on a semi-regular basis.
This is a great book filled with great proofs and just a great amount of information. Whenever I'm unsure of a proof or need confirmation of some obscure factoid of numbers, this is where I look first. Great for building the foundations needed for learning advanced mathematics.
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In this course, students will build upon the knowledge previously learned in Algebra I and Geometry, expanding their ability to understand, perform operations and solve problems with real numbers. Students will focus on the importance of linear relations; develop fluency with the language and operations of algebra to analyze and represent relationships; and perform exercises in statistics and statistical methods to solve problems.
Graduation Requirements:
This course can be used toward the Mathematics or the Electives requirement.
Course Materials:
Format
ISBN
Author
Title
Edition
Publisher
Notes
Online Textbook
Algebra II Materials
Must be ordered through TAA
**Materials will be sent automatically if you have not previously received them from us.
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Math 603-601 - Fall 2002
Summary
Definition - Vector space. A vector space is a set V
together with two operations, + and · . If u, v are in
V, then u + v is in V; if c is a scalar, then c·v is in
V. The operations satisfy the following rules.
Addition
Scalar multiplication
u + (v + w) = (u + v) + w
a·(b·u)
= (ab)·u
Identity: u + 0 = 0 + u = u
(a + b)·u = a· u +
b·u
Inverse: u + (-u) = (-u) + u = 0
a·(u + v) = a·u +
a·v
u + v = v + u
1·u = u
Subspaces
Definition - Subspace. A nonempty subset U of V is a
subspace if, under + and · from V, U is a vector space
in its own right.
Theorem. U is a subspace of V if and only if these hold:
0 is in U.
U is closed under + .
U is closed under · .
Example: U={(x1, x2, x3) |
2x1 + 3x2 - x3 = 0} is a subspace of
R3. On the other hand, W={(x1,
x2, x3) | 2x1 +3x2
- x3 = 1} is not a subspace of
R3, because 0 is not in W.
Important vector spaces
Displacements in space, forces, velocities, accelerations, etc.
+ parallelogram law
· usual scalar multiplication
Rn (real scalars) or Cn
(complex scalars) - n×1 real or complex matrices (i.e., columns;
can also work with rows).
Definition - Linear combination. Let v1 ...
vn be vectors in a vector space V. A vector of the form
c1v1 + ... + cn vn is
called a linear combination of the vj's.
Definition - Span. Let S={v1 ...
vn} be a subset of a vector space V. The span of S
is the set of linear combinations of vectors in S. That is,
U=span(S)={c1v1 + ... + cn
vn},
where the cjs are arbitrary scalars.
Definition - Linear independence and linear dependence. We
say that a set of vectors
S = {v1, v2, ... ,
vn}
is linearly independent (LI) if the equation
c1v1 + c2v2 + ... +
cnvn = 0
has only c1 = c2 = ... = cn = 0 as a
solution. If it has solutions different from this one, then the set S
is said to be linearly dependent (LD).
Inheritance properties of LI and LD sets. Every
subset of a linearly independent set is linearly
independent. Every set that contains a linearly dependent set
is linearly dependent.
Basis and dimension
Classification of vector spaces. If a vector space V has no
limit to the size of its linearly independent sets, it is said to be
infinite dimensional. Otherwise, V is said to be finite
dimensional. 2D and 3D displacements are finite dimensional
spaces. C[0,1] is infinite dimensional, because it contains {1, x,
x2, ..., xk } for all k.
When V is finite dimensional, its LI sets cannot be arbitrarily
large. Suppose that n is the maximum number of vectors an LI set in V
can have. A linearly independent set S in V having n vectors in it
will be called maximal.
Proof. Add any vector v in V to S to form the new set
{v, v1, v2, ... , vn}
This augmented set is LD, because it has n+1 > n vectors in
it. Thus we can find coefficients c, c1, c2,
... , cn, at least one of which is not 0, such
that
c v + c1v1 + ... + cn
vn = 0.
There are two possibilities. Either c is 0 or c is
not 0. In case c = 0, we have c1v1 + ... +
cnvn = 0. The set S is LI, so this
implies c1 = c2 = ... = cn = 0. But
this means all of the coefficients vanish, contradicting the fact that
one of them does not vanish. The only possibility left is
that c is not 0. We can then divide by it in our previous equation and
rewrite that equation as
v = (- c-1c1)v1 + ... + (-
c-1cn)vn .
This shows that S spans V. Our proof is complete.
Definition - Basis. We say that a set of vectors
S = {v1, v2, ... ,
vn}
in a vector space V is a basis for V if S is both
linearly independent and spans V.
Corollary. A linearly independent set
is a basis for a finite dimensional vector space if and only if it is
maximal.
Definition - Dimension. The dimension of a vector
space V is the maximum number of vectors in an LI set. Equivalently,
it is the number of vectors common to every basis for V.
Coordinates for a vector space. Assigning coordinates
amounts to providing a correspondence
v <-> (c1, c2, ... , cn)
that is 1:1, onto, and preserves vector addition and scalar
multiplication. Consider a basis {v1,
v2, ... , vn} for a vector space
V. We also assume that the basis is ordered, in the sense
that we keep track of which vector is first, or second, etc.
Theorem. Let B={v1, v2,
... , vn} be an ordered basis for a vector space
V. Every vector v in V can be written in one and only one way
as a linear combination of vectors in B. That is,
v =
c1v1 +...+
cnvn
where the coefficients are unique.
Proof. Since B is a basis, it spans V. Consequently, we have
scalars c1, ..., cn such that the representation
above holds. We want to show that this representation is unique -
i,e,, no other set of scalars can be used to represent
v. Keeping this in mind, suppose that we also have the
representation
v =
d1v1 +...+
dnvn,
where the coefficients are allowed to be different from the
ck's. Subtracting the two representations
for v yields
0 =
(c1 - d1) v1 +...+
(cn - dn) vn.
Now, B is a basis for V, and is therefore LI; the last equation
implies that c1 - d1 = 0, c2 -
d2 = 0, ..., cn - dn = 0. That is,
the c's and d's are the same, and so the representation is unique.
This theorem gives us a way to assign coordinates to V, for the
correspondence
v=
c1v1 +...+
cnvn
<-> (c1, c2, ... , cn)
it sets up is both 1:1 and onto. The condition of linear independence
gives us that it is 1:1, and the condition of spanning gives us that
it is onto. It is also easy to show that this correspondence preserves
addition and scalar multiplication, which are the properties needed in
defining "good" coordinates for a vector space.
Definition - Coordinate Vector. Given an ordered basis B =
{v1 ... vn} and a vector v
= c1v1 +...+
cnvn, we say that the column vector
[v]B = [c1, ...,
cn]T is the coordinate vector for
v, and that c1, c2, ..., cn
are the coordinates for v.
Definition - Isomorphism. Let U and V be vector spaces. A
correspondence between U and V
u <-> v
that is 1:1, onto, and preserves vector addition and scalar
multiplication is called an isomorphism between U and V, and
the two spaces are said to be isomorphic.
The word isomorphism comes from two Greek words, "isos," which means
"same", and "morphy," which means "form." As far as vector space
operations go, two isomorphic vector spaces have the "same form" and
behave the same way. Essentially, the spaces are the same thing, just
with different labels. For example,a basis in one space corresponds to
a basis in the other. Indeed, any property in one space that only
involves vector addition and scalar multiplication will hold in the
other. This makes the following theorem, which is a consequence of
what we said above concerning coordinates, very important.
Theorem. Every n-dimensional vector space is isomorphic to
Rn or Cn, depending on whether the
set of scalars is R or C.
Examples. The space P2 of polynomials of
degree 2 or less has B = { 1, x, x2 } as a basis, and so it
is three dimensional (B has three vectors). We take the scalars to be
real. Relative to this basis, we have
[p]B = [ a1 + a2x +
a3x2 ]B = [a1
a2 a3]T.
Thus, for example, we have these:
[1-x+x2]B = [1 -1 1]T
[-3x+4-2x2]B =
[4-3x-2x2]B = [4 -3 -2]T
[(2-x)2]B = [4-4x+x2]B =
[4 -4 1]T
Changing the ordering of the basis vectors changes the ordering of the
coordinates. Using the basis C = { x2 1, x }, we have
[-3x+4-2x2]C =
[-2x2+4-3x]C = [-2 4 -3]T,
which is a reordering of the coordinate vector relative to B. Be aware
that order matters!
This isn't the only isomorphism between P2 and
R3. Recall that a quadratic polynomial is determined
by its values at three distinct values of x; for instance, x=-1, 0,
and 1. Also, we are free to assign whatever values we please at these
points, and we can get a quadratic that passes through them. Thus, the
correspondence
p <-> [p(-1) p(0) p(1)]T
between P2 and R3 is both 1:1 and
onto. It is easy to show that it also preserves addition and scalar
multiplication, so it is another isomorphism between
P2 and R3. Let's use this to find
a new basis for P2. (Remember, a basis in one
isomorphic space corresponds to a basis in the other.) Since { [ 1 0 0
]T, [ 0 1 0 ]T, [ 0 0 1 ]T } is a
basis for R3, the set of polynomials
C = { p1(x) = -½x + ½x2,
p2(x) = 1, p3(x) = ½x +
½x2 },
which satisfy
p1(-1) = 1, p1(0) = 0, p1(1) = 0,
p2(-1) = 0, p2(0) = 1, p2(1) = 0,
p3(-1) = 0, p3(0) = 0, p1(1) = 1,
is another basis for P2. This raises the question of
how the coordinate vectors [p]C and [p]B are
related.
For purposes of comparison, we want to write out the expressions for
the coordinate changes. Writing the d's in terms of the b's, we have
dk = Ak1 b1 + ... +
Akn bn.
Going the other way, we can write the b's in terms of the d's,
bk = Ck1 d1 + ... +
Ckn dn.
We note that quantities transforming according to the formula for bases
are called covariant, and quantities transforming like the
coordinates are called contravariant.
The matrix that takes coordinates relative to B into ones relative to
D is A =
0 0 1
1/3 1/3 -1
-1/3 2/3 7
Solving a System of ODEs
Consider the following system of ordinary differential equations,
relative to x1-x2 coordinates.
dx1/dt = 3x1 + 2x2
dx2/dt = 2x1 + 3x2
We can turn this into a very simple decoupled system if we change from
x1-x2 coordinates to the
u1-u2 set defined via these equations:
x1 = u1 + u2
x2 = u1 - u2 .
Afrter some algebra, the system of ODEs becomes
du1/dt = 5u1
du2/dt = u2 ,
which can be easily solved.
Definition - Dual Space. The set V* of all
linear functions L:V - > R (or C) is called
the (algebraic) dual of V. The term linear means that
L(au + bv) = aL(u) +
bL(v)
holds for all scalars a,b and vectors u, v.
Terminology. Linear functions in the dual space are called
linear functionals, to distinguish them from other types of
linear functions. They are also called or 1-forms or
co-vectors.
Proposition. V* is a subspace of all
functions mapping V to the scalars.
Proof. We leave this as an exercise.
A simple physical example is the work W done by a force f
applied at a point and producing a displacement s. Here, the
work is given by W=L(s) =
f·s. The point is that is if we fix the
force, then the work is a linear function of the displacement. Note
that forces and displacements have different units and are
thus in different vector spaces, even though the spaces are
isomorphic.
Another simple example that frequently comes up is multiplication of a
column vector X by a row vector Y. The linear functional in this case
is just L(X) = Y X. Our final example concerns C[0,1]. L[f] =
0S1 f(x)dx is a
linear functional.
Dual Basis
Let V be an n-dimensional vector space. We want to construct a
basis for V*. Let B = {v1 ...
vn} be a basis for V. We may uniquely write any
v in V as
v = x1v1+ ...+ xnvn
Now, if L is a linear functional (i.e., it is in
V*), we also have
L(v) = x1L(v1)+ ...+
xnL(vn).
Thus knowing L(vj) for j=1 ... n completely
specifies what L(v) is. Conversely, given scalars
{y1, ..., yn}, one can show that
L(v) = x1y1 + ...+
xnyn,
where the xj 's are the components of relative to B,
defines a linear functional. As before, L(vj)
= yj. In summary, we have established this.
Theorem. Let V be a vector space with a basis B =
{v1 ... vn}. If L is a
linear functional in V*, then
L(v) = x1y1 + ...+
xnyn, where yj =
L(vj)
Conversely, given scalars {y1, ..., yn}, the
formula for L above defines a linear functional in
V*, where again L(vj) =
yj.
We can use the theorem we just obtained to define n linear functionals
{v1 ... vn} via
To make this clearer, let's look at what v1 does to
vectors. If we take a vector v = x1v1+ ...+ xnvn, then
v1(v) =
x1v1(v1) +
x2v1(v2) + ... +
xnv1(vn) =
x1·1 + x2·0 +
... +xn·0 = x1.
A similar calculation shows that v2(v) =
x2, v3(v) = x3, ...,
vn(v) = xn. This means that we can
write L(v) = x1y1 + ...+
xnyn as
L(v) = y1v1(v) + ... +
ynvn(v)
= (y1v1 +
... +ynvn)(v)
Now the two sides are equal for all values of the argument, so they
are the same function. That is, L =
yjv1 +...+
ynvn. Hence, the set B* =
{v1 ... vn} spans
V*. The set is also linearly independent. If 0 =
yjv1 +...+
ynvn, then
0=0(vj) = yj. Hence, the only
yj's that give 0 are all 0. Summarizing, we have
obtained this result.
Theorem. If V is an n-dimensional vector space, and if B =
{v1 ... vn} is a basis for V,
then the dual space V* is also n-dimensional and
B* = {v1 ... vn} is a
basis for V*.
Definition - Dual Basis. The basis B* is called
the dual basis for B.
Inner Product
Definition - Inner product Let V be a real vector
space. We say that a mapping < , > : V×V --> R is
an inner product for V if these hold:
positivity - <v,v> > 0, with
<v,v> = 0 implying that v=0.
symmetry - <u,v> =
<v,u>
homogeneity - <cu,v > =
c<u,v >
additivity - < u+v,w> =
<u,w> + <v,w>
Definition - Norm The quantity ||v|| :=
(<v,v>)½ is called the
norm or length of a vector v.
Definition - Inner product space A vector space together
with an inner product defined on it is called an inner product
space.
Examples We verified in detail that the following are inner
products on the spaces listed. In particular, we motivated the
selecting the inner product on C[a,b] by working with the one for
Rn, modifying it, and letting n tend to infinity.
Orthogonal vectors. From the definition of the angle between two vectors, we can
rewrite an inner product in a form familiar from the definition of the
"dot product" of vectors in 2D and 3D. If we let t be the angle
between u and v, then
<u,v> = ||u|| ||v|| cos(t),
If neither of the vectors are 0, then t = π/2 if and only if
the inner product on the left is 0. With this in mind, we say that two
vectors u,v in an inner product space V are
orthogonal or perpendicular whenever
<u,v> = 0.
Things are even simpler in an orthonormal basis. If we let B =
{u1, u2, u3,
..., un} be orthonormal, then ||
uj || = 1, and
xj =
<v,uj>.
This is familiar from 3D vectors, with {i, j,
k} being the orthonormal basis.
We now want to address what happens if we change from B to a new
orthonormal basis, B' = {u'1, u'2,
u'3, ..., u'n}.
u'j=A1ju1 +
A2ju2 + ... +
Anjun.
If the matrix A has j,k entry Akj, then the
coordinate vectors transform according to the rule
[v]B = A[v]B'. By our previous
proposition, we thus have that
<v,w> = [w]BT
[v]B = [w]B'T AT
A[v]B'
On the other hand, the proposition applies directly to the
basis B' itself. Hence,
<v,w> = [w]B'T
[v]B'. Combining these two equations then gives us
[w]B'T
[v]B' = [w]B'T AT
A[v]B',
which holds for any choice of vectors v and w.
We are interested in getting the components of ATA. To do
this, choose w = u'j and v =
u'k. The coordinate vectors for these are
[w]B' = [u'j]B' =
ej and [v]B' =
[u'k]B' = ek. Inserting
these in the equation above gives us
ejT ek =
ejT ATA ek
This implies that the (j,k) entry in ATA is 1 if j=k and 0
if j is not equal to k. But these are exactly the entries in the
n×n identity matrix I. Thus, we have shown that ATA =
I.
Definition - Orthogonal matrix An n×n matrix A is
said to be orthogonal if ATA = I.
Proposition. The following are equivalent.
ATA = I
AT = A-1
The columns of A form an orthonormal basis for
Rn
The rows of A form orthonormal basis for
Rn.
In Rn, the length of Ax is the same as x, and
the angle between Ax and Ay is the same as that between x and y.
Euler angles
Rotations and reflections. A change from one orthonormal
basis to another is accomplished by an orthogonal matrix. This change
of basis leaves lengths and angles invariant, and in 3D represents
a rotation or reflection. For a 3×3 real matrix, being
orthogonal implies six equations for the nine
entries in A. To parametrize the 3×3 orthogonal matrices should
then require three variables. These variables are the Euler
angles, and they come from three rotations. The angles are called the
precession, nutation, and pure rotation. A
diagram may be found in the book by Borisenko and Tarapov.
Rotation about the z-axis Here, the z-axis is fixed, and
we want to rotate our axes counterclockwise though an angle t. Let B =
{i, j, k}, and B' = {i', j',
k'}. The relationship between these two bases is
QR-Factorization Suppose m>=n. Let A be an m×n
matrix with linearly independent columns {v1, ...,
vn}; that is,
A = [v1 ... vn].
Apply the Gram-Schmidt process to the vj's. The
result is that for k = 1, ..., n, we have
vk
= r1ku1 + ... +
rkkuk,
where the uj's are also column vectors. Put the
uj's in an m×n matrix
Q = [u1 ... un].
Letting R =
we can write the equations that give vk's as linear
combinations of uj's in matrix form, A = QR. This is
the QR factorization. Whem m = n, the matrix Q is orthogonal. If m
> n, then Q satisfies QTQ =
In×n. However, QQT will not be
the m×m identity Im×m
Discrete least squares problems The simplest example of
this is fitting a straight line to data. For instance, suppose we have
measured the log of the concentration of some chemical and listed the
data in the table below.
Log of Concentration
t
0
1
2
3
4
ln(C)
-0.1
-0.4
-0.8
-1.1
-1.5
We know that the law of decay tells us that ln(C) = -r*t +
ln(C0), so the the data should lie on a straight line. Of
course, they don't; experimental errors offset the points. The
question is, what are values for r and ln(C0) that will fit
a straight line to the data? A more general problem is this. At times
t1, t2, ..., tn, we have measurements
y1, y2, ..., yn. Find a line y = a*t +
b that fits the data.
One good way to solve this problem is the method of least squares. Let
E2 = (|y1 - a*t1 -b|2 +
|y2 - a*t2 -b|2 + ... +
|yn - a*tn -b|2)/n
The quantity E is the root mean square of all of the errors
yj - a*tj -b at each time tj. The
idea is to choose a and b so as to minimize E. That is, we will try to
find a and b that give the least value for the sum of the
squares of the errors. We can put this in the form of an
inner product. Define the following column vectors in
Rn.
v = [y1 y2 ...
yn]T v1 = [t1 t2 ...
tn]T v2 = [1 1 ... 1]T
Notice that the jth component of the vector
v - av1 - bv2 is just the
difference yj - a*tj -b. Form this it follows
that
E2 = ||v - av1 -
bv2||2/n
Now, n is simply the number of data points, so it is
fixed. Consequently, minimizing E then is equivalent to minimizing the
the distance from v to the space spanned by
v1 and v2. Put a little
differently, minimizing E is equivalent to finding the
v* in span{v1,
v2} that comes closest to v or best
approximatesv.
Continuous least squares problems There is a continuous
version of the discrete problem described above. In many applications,
we are given a complicated function and we want to approximate it with
sums of simpler functions. A familiar example is the Taylor series for
ex, where we are approximating ex by a sum of
powers of x. Another example is using sines and cosines to approximate
a signal in order to find its frequency content. This is one of the
applications of Fourier series.
Suppose that we are given a continuous function f(x) on the interval
[0,1] that has an upward trend or bias to it. One way to
measure this is to fit a straight line to the function f. The
difference here is that we know f at every x in [0,1]. The
discrete square error E2 goes over to an integral,
E2 = 0S 1
(f(x) - ax -b)2dx .
If we use the inner product < f,g > = 0S 1 f(x)g(x)dx, then E2 = ||
f(x) - ax -b||2, and the problem again goes over to finding
the best approximation to f from the span{1,x}, relative to
the norm from our inner product < f,g >.
One can carry this further. If f(x) has not only an upward trend, but
is also concave up, then it makes sense to fit a quadratic to it. The
problem described above would change to finding the quadratic
polynomial f*(x) in span{1,x,x2} that minimizes || f(x) -
a0 - a1x - a2x2
||2.
Least squares problems and inner products All of the
problems that we have described above have been put in terms of inner
products. Here is the general form of these problems. Suppose that we
have an inner product space V, a vector v in V, and a subspace
U of V. The least-squares problem is to find both the minimum
of || v - u ||, where u is any vector in U, as
well as any minimizer v* in U.
Normal equations
Theorem. Let V be a vector space with an inner product <
u, v >, and let U be a subspace of V. A vector v* in U
minimizes the distance || v - u || if and only if
v* satisfies the normal equations,
< v - v*, u > = 0,
which hold for all u in U. That is, v - v* is
orthogonal to the whole space U. In addition, v* is
unique.
Proof. Let's first show that if v* in U minimizes ||
v - u ||, then it satisfies the normal equations. The
way we do this is similar to the way we proved Schwarz's
inequality. Fix u in U and define
q(t) := || v - v* + t u) ||2 = ||
v - v* ||2 + 2t < v - v*,
u > + t2 || u ||2
Because v* minimizes || v - u ||2 over
all u in U, the minimum of q(t) is at t = 0. This means that t
= 0 is a critical point for q(t), so q'(0) = 0. Calulating q'(0) then
gives us 2< v - v*, u > = 0 for all
u in U. Dividing by 2 yields the normal equations.
Conversely, if v* in U satisfies < v - v*,
u > = 0, then we will show not only that v* is
a minimizer, but also that it is the minimizer; that
is, v* is unique. To do this, let u be any vector in
U. Observe that we can write v - u = v -
v* + v* - u = v - v* + u',
where u' := v* - u is in U. Consequently, we also
have
|| v - u ||2 = ||v - v* +
u' ||2
|| v - u ||2 = ||v - v*
||2 + 2< v - v*, u' > + ||
u' ||2.
Since we are assuming that v - v* is orthogonal to every
vector in U, it is orthogonal to u'; hence, < v -
v*, u' > =0, and so have that
|| v - u ||2 = ||v - v*
||2 + || u' ||2.
It follows that || v - u || >= ||v - v*
||, so that v* is a minimizer. Now, if equality holds,
that is, if || v - u || = ||v - v* ||,
then we also have || u' || = 0. Consequently, u' =
0. But then, we have to have u = v*. So, the
vector v* is unique.
Normal equations relative to a basis. The normal equations
are geometric conditions that can be used to directly find the
minimizer. When the subspace U has dimension n, they reduce to a set
of n equations involving basis vectors.
Corollary If B = {w1
... wn} is a basis for the subspace U, then the
normal equations are equivalent to the set
< v - v*, wk > = 0, k = 1 ... n.
Proof. If the normal equations are satisfied, they hold for
every vector in U, including the basis vectors. Thus, the equations
above have to hold, too. On the other hand, suppose the equations
above are satisfied. We can write any vector u in U as u
= c1w1+ ... +
cnwn. It then follows from the equations
above that
< v - v*, u > = c1 < v
- v*,w1 > + ... + cn <
v - v*,wn > = c1·0 +
... + cn·0 = 0,
and so the normal equations hold.
Finding the minimizer - orthonormal
case. The normal equations are geometric conditions that can
be used to directly find the minimizer v*. When an orthonormal
basis is for U is known, the answer is simple, and can be explicitly
written down. We just need to apply the last corollary. Take B =
{u1 ... un} to be an orthonormal
basis for the subspace U. The normal equations relative to this basis
are < v - v*, uk > = 0 or,
equivalently,
< v, uk > = <v*,
uk >,
for k=1 .. n. By the formula the coordinates of a vector
relative to an orthonormal basis, we see that <v*,
uk > is just the kth coordinate of
v*. It follows that the minimizer is given by
v* = < v , u1 >
u1+ ... + < v , un >
un
It's worth noting that this is the first time we have actually shown
that there is a minimizer. Of course, by what we have said
above, it's unique. We will use this fact later.
An Example. Suppose that f(x) = ex on
[-1,1]. What straight line gives the best least squares fit? The first
thing to do is identify the subspace. Here U =span{1,x}. We know that
{2-½, (3/2)½ x} form an
orthonormal basis for U relative to the inner product
Here, we interpret p and q as continuous functions, rather than
polynomials. Doing a little calculus, we obtain
< ex, 2-½ > =
2-½(e - e-1) =
2½sinh(1)
< ex, (3/2)½ x > = 6½
e-1.
Applying the formula we derived, we get f*(x) =
2½sinh(1)·2-½ +
6½ e-1·(3/2)½ x
= sinh(1) + 3e-1 x. The function and the line are plotted
below.
Finding the minimizer - non-orthogonal case. Often, it is
useful to use a non-orthogonal basis in solving a least squares
problem. Let B = {w1 ... wn} be a
basis for the subspace U. We showed that the normal equations, which
determine the minimizer v*, have the form
< v - v*, wk > = 0, k = 1 ... n.
Since v* is in U, we can represent it in terms of this basis,
v* = c1w1+ ... +
cnwn. The normal equations imply that the
coefficients cj satisfy the matrix equation
Gc=d, where Gjk= < wk,
wj >, dj = < v ,
wj >.
The matrix G is called the Gram
matrix for the basis of w's; it is always invertible,
because the normal equations always have a unique solution,
as we saw above in connection with the case of an orthonormal basis.
The normal equations and the QR factorization Suppose that
V is Rn and U has a basis B = {w1,
..., wm} of m column vectors, where m lt; n. Given
v in V, We want to minimize ||v - u|| over all
u in U. As we have seen, the unique minimizer
v* = c1w1+ ... +
cnwn.
Instead of employing the Gram matrix, as we did earlier, we use our
"basic matrix trick," and write v* as a matrix product,
v* = Wc,
where W = [w1 ... wm] is an
n×m matrix with linearly independent columns. Now, carry out a
QR factorization and write W = QR, where R is an invertible m×m
upper triangular matrix and where the columns of Q are form an
orthonormal set {u1 = Qe1, ...,
um = Qem} that is also a basis for
U. The normal equations, relative to the basis comprising the columns
of Q, become, for k = 1, ..., n,
Least squares approximation Consider the following
continuous least squares problem. Start with a function f defined on
[-1,1], which we can think of as continuous, and suppose that we want
to fit not only a straight line to f, but also a quadratic, cubic,
quartic, and so on. In other words, we want to find the degree n
polynomial that gives the best least squares fit to the function f
over the interval [-1,1].
The orthonormal set of Legendre polynomials is formed by using the
Gram-Schmidt process on {1, x, x2, x3, ...}
relative to the inner product
We denote these polynomials by {p0, p1,
p2, p3, ...}. Earlier, we had seen that
p0 = 2-½, p1 =
(3/2)½ x, p2 = (5/8)½
(3x2 - 1).
We remark that there are similar formulas for all of the orthonormal
Legendre polynomials.
For each n, we look at the subspace Un =
span{p0, p1, p2, ...,
pn}. Of course, Un = Pn, the
polynomials of degree n or less. In our least squares minimization
problem, we identify f with v and the uk's with
the pk's. The minimizer for Un is
f*n = < f, p0 >
p0+ ... + < f, pn >
pn.
The minimizers f*n change with n in a very simple
way. Namely, to go from n to n+1, we only need to add a term to the
previous minimizer. If we formally let n tend to infinity, then we get
the infinite series
< f, p0 >
p0 + < f, p1 >
p1 + < f, p2 >
p2 + < f, p3 > p3 + ...
for which the minimizer f*n is the nth partial
sum.
Let the minimum error over Un be En = || f -
f*n ||. Because Un = Pn
is contained in Un+1 = Pn+1, we must have
En+1 <= En. That is, En decreases
as n gets bigger. Does the En go to 0 as n -> infinity?
If it does, we say that f*nconverges in the
mean. We also say that the series converges in the mean to f, and
we also write
f = < f, p0 >
p0 + < f, p1 >
p1 + < f, p2 >
p2 + < f, p3 > p3 + ... .
Theorem The series above converges in the mean to f if and
only if
|| f ||2 = |< f, p0 >|2 + |<
f, p1 >|2 + |< f, p2
>|2 + |< f, p3 >|2 + ...
This formula is called Parseval's equation, and first
appeared in connection with Fourier series.
Orthonormal series What we just said applies to any
infinite set of orthonormal functions, including the trigonometric
functions
Both Fourier series and series of Legendre polynomials converge in the
mean whenever the square of the appropriate norm, || f ||2,
is finite.
Fourier series. The Fourier series for a
2 periodic function f is usually written
as
where the coefficients are given by
As an example, we calculated the Fourier series for the 2 periodic extension of the function |x| defined on the
interval [-,]. The resulting series was
½ - 4-1 (cos(x) + 3-2cos(3x) +
5-2cos(5x) + 7-2cos(7x) + ...)
We used MATLAB to do several examples of finding the best
continuous least-squares fit using Legendre polynomials. We also
looked at the partial sums of the Fourier series for the 2 periodic extension of |x|.
Linear transformations
Definition A mapping L:V -> W, where V, W are vector
spaces is said to be a linear transformation if it satisfies
these properties.
Let V and W be finite dimensional, and let B =
{v1, ... , vn} and D =
{w1, ... , wm} be bases for V and W,
respectively. If L:V -> W is a linear transformation, then there is
a unique m×n matrix A such that w = L[v] holds if
and only if A[v]B = [w]D.
To find A, we first find the output of L applied to each basis vector;
that is, L[1], L[x], and L[x2]. Doing this, we obtain L[1]
= 3, L[x] = 1 + 5x, and L[x2] = 2x + 9x2. By the
construction in the theorem, the kth column of A is the coordinate
vector [ L[vk] ]D. Consequently, we have
[ L[1] ]D = [3 0 0]T
[ L[x] ]D = [1 5 0]T
[ L[x2] ]D = [0 2 9]T
We have now found that the matrix A =
Co-domain. The vector space W is called the
co-domain. (This is sometimes called the range.)
Null space. The set of all vectors null(L) := {v in V:
L[v] = 0}; it can be shown to be a subspace of V.
Image. The set of all vectors image(L) := {w in W:
w = L[v] for somev in V}; it can be
shown to be a subspace of W. (This set is also sometimes called the
range.)
Examples
ODEs Consider the ODE y"+p(x)y'+q(x)y = g(x). We convert
this to operator form L[y]=g, where L[y] = y"+p(x)y'+q(x)y. In this
case, the domain V is C(2)(R), the space of twice
continuously differentiable functions on R and the co-domain W
is C(R), the space of continuous functions on R. The
null space, null(L), is the set of solutions to the homogeneous
equations. With a little work, one can show that image(L) is also
C(R).
Matrices Consider the 2×4 matrix M =
1 -1 2 3
1 1 1 4
We let L be the transformation L[v] = Mv, where v
is in R4. For this problem, V =
R4. The co-domain or set of outputs W is
R2. Here, we also have image(L) =
R2. The null space is the set of all v for
which Mv = 0. Using row-reduction, one can show that
null(L) = span{[-3 1 2 0]T, [-7 -1 0 2]T}
Combinations of linear transformations
Sums K+L is defined by (K+L)[v] =
K[v] + L[v].
Scalar multiples cL is defined by (cL)[v] = c(L[v]).
Products If K : V -> U and L : U -> W are linear, then
we define LK via LK[v] = L[K[v]]. (This is composition of
functions). The transformation defined this way, LK, is linear, and
maps V -> W. Note: LK is not in general equal to KL, which
may not even be defined.
Inverses Let L : V -> V be linear. As a function, if L
is both one-to-one and onto, then it has an inverse K V -> V. One
can show that K is linear, and LK = KL = I, the identity
transformation. We write K = L-1.
Associated matrices Recall if B = {v1,
... , vn} and D = {w1, ... ,
wm} be bases for V and W, respectively, then the matrix
associated with the linear transformation L : V -> W is
ML = [ [ L[v1] ]D, ... , [
L[vn] ]D ]
Since each of the combinations listed above is still a linear
transformation, it will have a matrix associated with it. Here is how
the various matrices are related.
MK + L = MK + ML
McL = c ML
MLK = ML MK
ML-1 = (ML)-1
Polynomials in L : V -> V We define powers of L in the
usual way: L2 = LL, L3 = LLL, and so on. A
polynomial in L is then the transformation
p(L) = a0I + a1L + ... + amLm
Later on we will encounter the Cayley-Hamilton theorem, which says
that if V has dimension n, then there is a degree n (or less)
polynomial p for which p(L) is the 0 transformation.
Change-of-basis
Change-of-basis and linear
transformations Let L : V -> V be linear. Here L maps V
into itself. We want to look at what happens when to the matrix of L
when we make a change of basis in V. Let V have bases
B = {v1, ... , vn}
and B' = {v'1, ... , v'n}
If the matrix of L relative to B is ML, and that relative
to B' is M'L, then
M'L = SB -> B' ML (SB ->
B')-1, where SB -> B' changes B
coordinates to B' coordinates.
Eigenvalue problems We say that a scalar µ is an
eigenvalue of L : V -> V if there is exists a vector
v in V, with v not equal to 0, such that
L[v] = µ v. The vector v is called an
eigenvector of µ. We let Eµ be
set eigenvectors associated with µ, along with
0. Eµ is a subspace of V that is called
the eigenspace of µ.
Trajectories of a velocity field Suppose that we are given
a time-independent velocity field for a 2D fluid,
V(x). Assume that V is a linear function of
x, so that V(x) = L[x], where L is a
linear transformation taking 2D to 2D. We want to find the
trajectories of the fluid particles, dx/dt = V(x)
= L[x], given that we know the initial position
x(0). Let B = {i,j} be the usual basis for
2D. Relative to this basis L will have a matrix A. so our problem
becomes [dx/dt]B = A[x]B. Since
the basis B is time-independent, [dx/dt]B =
d[x]B/dt. If we let X = X(t) =
[x]B=[x1 x2]T, we
arrive at the system
dX/dt = AX, where X(0)= [x(0)]B.
To make the situation more concrete, we will assume that A =
1 4
1 -2
When this system is written out for this choice of A, it looks like
dx1/dt = x1 + 4x2
dx2/dt = x1 - 2x2
The idea here is to switch to a basis where the system decouples. To
accomplish this, we will use a basis of eigenvectors for A,
{[4 1]T, [-1 1]T}. These correspond to the
eigenvalues of A µ = 2 and µ = -3, respectively. (We will
discuss how to get these later). Relative to our original 2D space, we
write this basis as B' = {i+4j, -i+j}. The
change of basis matrix S = SB'
->B is given by S =
Letting Z = [x]B' = [z1
z2]T, we have the new system dZ/dt = A'Z, with
Z(0) = [x(0)]B'. In the new coordinates the system
decouples and becomes
dz1/dt = 2z1
dz2/dt = - 3z2
This decoupled system is easy to solve. The result is
z1(t) = e2tz1(0)
z2(t) = e-3tz2(0)
Transforming back to the original coordinates, we have that X =
SD(t)S-1X(0), where D(t) =
exp(2t) 0
0 exp(-3t)
This explicitly solves the problem. However, we still have to explain
how to find the eigenvalues and eigenvectors of A.
Finding eigenvalues and eigenvectors Let A be an n×
n matrix, and define pA(µ) = det(A - µ I). One
can show that pA is a polynomial of degree n. The scalar
µ is an eigenvalue of A if and only if it is a root of
pA. This follows from two obeservations. First, µ is
an eigenvalue of A if and only if for some x not equal to 0 Ax
= µ x; this in turn is equivalent to (A - µ I)x = 0, so A
- µ I is singular. Second, A - µ I is singular if and only
if det(A - µ I) = 0.
For A, the problem of finding eigenvalues and eigenvectors
decouples. One can first find the roots of pA(µ),
and then solve a linear system to get the eigenvectors.
If the scalars are the complex numbers, then the Fundamental
Theorem of Algebra implies that pA has at least one
root. Consequently, A has at least one eigenvalue.
To find the eigenvalues and eigenvectors of a linear
transformation L, choose a basis for V and find the matrix of L
relative to this basis, ML. The eigenvalues of L are
precisely those of ML; the eigenvectors of L have
coordinates corresponding to the eigenvectors of ML.
which has X = x2·[4 1]T as a
solution. Repeating the argument for µ = -3 results in X =
x2·[-1 1]T. To get the basis we want, we
choose x2 = 1 in both cases. Other nonzero values will work
equally well.
Diagonlizable linear transforms A linear transformation T :
V -> V is diagonalizable if and only if there is a basis B
for V comprising only eigenvectors of L. When the happens
ML, the matrix of L relative to the basis B will be a
diagonal matrix. Conversely, if there is a basis relative to which the
matrix of L, ML, is diagonal, then that basis will be
composed of eigenvectors.
This matrix is not diagonalizable. It's only eigenvalue is
µ=1 and the only eigenvectors have the form c·[1
0]T, where c is a scalar. There is no second linearly
independent eigenvector, and so there is no basis of eigenvectors.
Applications
Normal modes of a spring system We consider a horizontal
spring system consisting of three identical springs (constant = k) and
two identical masses (mass = m), all attached to wall mounts. The
displacements from rest of the first and second masses are
x1 and x2, respectively. Newton's laws for the
system give these equations of motion
md2x1/dt2 = -2kx1 +
kx2
md2x2/dt2 = kx1 -
2kx2
We can put these equations in matrix form. Let X = [x1
x2]T. The equations become the single matrix
equation
md2X/dt2 = - kAX, where the matrix A =
2 -1
-1 2
A normal mode of this system is a solution of the form X(t) =
Xweiwt, where Xw is independent of t
and not equal to 0. Plugging this solution back into
the matrix equation yields, after cancelling eiwt,
AXw = (mw2/k) Xw
We see that Xw is an eigenvector of A corresponding to the
eigenvalue µ = mw2/k. Thus we need to find the
eigenvalues and eigenvectors of A. As usual, we obtain the
characteristic polynomial:
pA(µ) = det(A - µI) = (2 -
µ)2+1.
The roots are µ = 1 and µ = 3, and the corresponding
eigenvectors are [1 1]T and [-1 1]T. In writing
the normal modes, we use both e±iwt or,
equivalently, cos(wt) and sin(wt).
[1 1]Texp(±i(k/m)½t)
frequency = (k/m)½
[-1 1]Texp(±i(3k/m)½t)
frequency = (3k/m)½
Physically, in the lower frequency mode both masses move in tandem
together, -> -> or <- <-. In the higher frequency mode,
they move exactly opposite each other, -> <- or <- ->.
A circuit We want to analyze the circuit shown below. In
the circuit, E(t) is a voltage source, R1 and R2 are resistors, L is a
coil, and C is a capacitor. The state variables for this system are
I=IL, the current through L, and V = VC, the
voltage drop across C.
The characteristic polynomial for this matrix is
pA(µ) = (-1 -µ)2 + 1. The two
eigenvalues are then
µ± = -1 ± i.
The augmented matrix representing the system (A
-µ+I)X = 0 is
-i 1 0
-1 -i 0
This is equivalent to the single equation -ix1 +
x2 = 0. Up to nonzero multiples, the eigenvector for
µ+ is [1 i]T. Either by repeating these
steps with µ- or by taking complex conjugates, we
have that the eigenvector for µ- is [1
-i]T. As in our previous examples, we form the
change-of-basis matrix S = SB' -> B =
Similar matrices Recall that when we change
coordinates, the two matrices representing a linear transformation L
are related by M'L = S-1MLS. We say
that an n×n matrix B is similar to an n×n matrix
A if there exists an invertible matrix S such that B =
S-1AS. Simple properties of similarity are
listed below. We remark that these properties together make similarity
an equivalence relation.
B is similar to A if and only if A is similar to B. (Thus, we can
simply say that A and B are similar matrices.)
A is similar to A.
If A is similar to B, and B is similar to C, then A is similar to
C.
Diagonalizable matrix A linear transformation L was defined
to be diagonalizable if there was a basis relative to which its matrix
ML was diagonal. If we regard an n×n matrix A as a
linear transformation, then the condition for it to be diagonalizable
is that there is a matrix S such that S-1AS is a diagonal
matrix. Equivalently, A is diagonalizable if and only if it is similar
to a diagonal matrix.
Determinants - A Quick Tour
Permutations Permutations of the integers 1 through n are
either even or odd. A permutation is even
if it can be achieved by an even number of interchanges
(transpositions), and odd if it takes an odd number of
them. There are n! permutations. We define the function sgn(p) to be
+1 if p is an even permutation, and -1 if p is odd.
Definition of a determinant If A is an n×n matrix,
then we define det(A) via
det(A) = SUMp sgn(p) ai1,1ai2,2
...ain,n , p = (i1,i2,...,in)
Basic properties of determinants These properties follow
immediately from the definition. On the other hand, they characterize
the determinant. Only det(A) satisfies them.
Multilinearity. Let A
=[a1,a2,...,an]. Then
det([a1,a2,...,an]) is a linear
function of column ak, if all other columns are held fixed.
Alternating function. Interchanging two columns changes
the sign of the determinant.
Degree The degree of the characteristic polynomial for an
n×n matrix is precisely n.
Similar matrices If B = S-1AS, then
pB(t) = pA(t).
Necessary conditions for a matrix to be diagonalizable
Theorem. If an n×n matrix A has n distinct
eigenvalues, then A is diagonalizable.
Proof. Let µ1, µ2,...,
µn be the n distinct (possibly complex) eigenvalues
of A; equivalently, these are the roots of pA. Let
v1, v2,..., vn. Be n eigenvectors
corresponding to these n eigenvalues. We will show that these
eigenvecotrs form a linearly
independent set of vectors. We start with the equation
c1v1 + c2v2 + ... +
cnvn = 0
Consider the matrix P1 = (µ2I - A)(
µ3I- A)...(µnI - A). Using the fact
that the vk's are eigenvectors of A, we have that
P1vk = 0 if k=2,3,...,n
P1v1 = (µ2 -
µ1)( µ3-
µ1)...(µn -
µ1)v1,
so applying P1 to both sides of the previous equation
results is
c1(µ2 - µ1)(
µ3- µ1)...(µn -
µ1)v1 = 0
Since v1 is not0, and since the
eigenvalues are distinct, we have that c1=0. Repeating this
procedure with the remaining coefficients implies that all of the c's
are 0, and so the set is a basis because it is a maximal linearly independent
set.
Selfadjoint matrices
Adjoints The adjoint of a linear
transformation L : V -> V, where V is an inner product space, is
the unique linear transformation L' that satisfies
< L'[u],v > = < u, L[v] > .
For a real m×n matrix A, the adjoint is the transpose
AT. If the matrix is complex, then the adjoint of A is
AH, the conjugate transpose. A matrix is
selfadjoint or Hermitian if it is equal to its own
adjoint. Thus, if A is real, A = AT; i.e., A is symmetric.
Properties of selfadjoint matrices
The eigenvalues of a selfadjoint matrix are real.
Eigenvectors corresponding to distinct eigenvalues are
orthogonal.
Every selfadjoint matrix is diagonalizable.
Every selfadjoint matrix has an orthonormal basis
relative to which it is diagonal. Equivalently, for a real n×n
matrix, there is an orthogonal matrix S such that
STA S is a diagonal matrix.
We wish to find coordinates relative to which the "cross terms" are
removed. A set of axes that has this property is called
principal. Since A is selfadjoint, it can be diagonalized by
an orthogonal transformation S. That is, there is a diagonal matrix D
such that D = STAS. Because S is orthogonal, we also have A
= SDST. Now, let Y = [u v w]T =
STX. We see that
XTAX = YTDY = µ1u2 +
µ2v2 + µ3w2,
and the cross terms have been transformed away by rotating and
(possibly) reflecting coordinates.
We now will finish the probelm by finding the eigenvalues and
eigenvectos of A. The characteristic polynomial of A is
pA(µ) = det(A - µI) =
5-µ 1 1
1 5-µ 1
1 1 5-µ
We do not change the determinant by subtracting (5-µ)×
column 2 from column 1. Thus, pA(µ) =
0 1
1
1-(5-µ)2 5-µ 1
1-(5-µ) 1 5-µ
We can factor 1-(5-µ) = µ-4 out of the first column. The
result, after simplifying, is that pA(µ) =
(µ-4)×
0 1
1
6+µ 5-µ 1
1 1 5-µ
We now subtract the last column from the second. We obtain
pA(µ) = (µ-4)×
0 0
1
6+µ 4-µ 1
1 µ-4 5-µ
Next, we factor µ-4 out of the second column to get
pA(µ) = (µ-4)2×
0 0
1
6+µ 1 1
1 -1 5-µ
Evaluating the last 3×3 determinant is easily done. The final
result is
pA(µ) = (µ-4)2(7-µ)
Now we will find the eiegnvectors correspond to µ=4 and
µ=7. For µ=4, the augmented matrix [A-4I|0] is
1 1 1 0
1 1 1 0
1 1 1 0
This system is equivalent to the single equation x+y+z=0. There are
two linearly independent solutions:
[-1 1 0]T and [-1 0 1]T.
Any linear combination of these is still an eigenvector. Using the
Gram-Schmidt process, we
can convert this to an orthonomal set,
2-½[-1 1 0]T and 6-½[-1
-1 2]T
The eigenvector for µ=7 is found by solving the system with
augmented matrix [A-7I|0]
-2 1 1 0
1 -2 1 0
1 1 -2 0
The resulting eigenvector is 3-½[1 1
1]T. Finally, S =
-2-½ -6-½
3-½
2-½ -6-½
3-½
0 2·6-½
3-½
Relative to the new coordinates, the quadratic form becomes
YTDY = 4u2+4v2+7w2.
Motivation We return to the circuit we
were working with earlier. (See the class notes for October 22.) Assume that the various
components have these values:
E(t) = 0, L =1 henry, C = 1 farad, R1 = 1 ohm,
R2 = 3 ohm.
The matrix equation for the circuit is dX/dt = AX + F(t), where X = [I
V]T. With these values for the components, F(t) = 0,
and
A =
-3 1
-1 -1
The characteristic polynomial for this matrix is
pA(µ) = µ2 + 4µ + 4 =
(µ+2)2. Thus, there is only one eigenvalue, µ =
-2. The augmented matrix representing the system (A -µI)X = 0 is
-1 1 0
-1 1 0
This is equivalent to the single equation -x1 +
x2 = 0. Hence, up to nonzero multiples, the eigenvector for
µ = -2 is [1 1]T. There are no other linearly
independent eigenvectors, and so A is not
diagonalizable. What this means is that the system of equations
doesn't completely decouple. It does partially decouple,
though. Consider a new basis B' = {[1 1]T, [0
1]T}. As before, let S = SB'->B =
1 0
1 1
Setting Z = SX, the system becomes dZ/dt = S-1ASZ. Doing a
little matrix algebra, we see that S-1AS =
-2 1
0 -2
This is called the Jordan normal (canonical) form of A. The
new system is
dz1/dt = -2z1 + z2
dz2/dt = -2z2
Solving the second equation gives us z2(t) =
e-2t z2(0). Substituting this into the first
equation results in a nonhomogeneous, linear, first order differential
equation,
dz1/dt = -2z1 + e-2t z2(0).
We can solve this using integrating factors. (See an ODE text for an
explanation). The solution is
z1(t) = e-2t z1(0) + te-2t
z2(0)
Finally, X(t) = S-1T(t)SX(0), where T(t) =
e-2t te-2t
0 e-2t
Block upper triangular form Every matrix A has a basis
relative to which it is in block triangular form. This means that we
can find an invertible matrix S such that S-1AS =
T1,1
0
0
...
0
0
T2,2
0
...
0
...
...
...
...
...
0
0
0
...
Tr,r
Each diagonal block Tk,k is upper triangular, with the
diagonal entries all being an eigenvalue µk repeated
as many times as it is a root of the characteristic polynomial. For
example, if µ7 is repeated four times, then
T7,7 =
µ7
*
*
*
0
µ7
*
*
0
0
µ7
*
0
0
0
µ7
Jordan normal (canonical) form Every matrix A has a
basis relative to which the blocks Tk,k are Jordan
blocks, Jm(µ). This is an m×m matrix with
µ's down the diagonal, 1's down the superdiagonal, and 0's
elsewhere. For example, if m = 6 and µ = 3, then
J6(3) =
3
1
0
0
0
0
0
3
1
0
0
0
0
0
3
1
0
0
0
0
0
0
3
1
0
0
0
0
0
3
Two matrices having the same Jordan normal form, apart from ordering of the
blocks along the diagonal, are similar. An m×m matrix A is similar
to Jm(µ) if and only there is a basis
{f1, ..., fm}
satisfying
Invariance Physical laws should be formulated in ways that
are independent of any coordinates used; that is, the laws should be
stated in such a way that they are invariant under transformation of
coordinates. For scalar quantities this has a simple meaning. If we
know the temperature on a surface T(P) in u-coordinates, so that
T=f(x). Then in another coordinate system, x =
g(x'), the temperature is given by T=f(g(x')). The law of
transformation is simply composition of functions (substitution).
Vectors Vectors are displacements in ordinary three
dimensional space, or physical quantities, like velocity, acceleration,
force, and so on, that can be represented by such
displacements. Invariance means describing these vectors in
a way that is independent of the choice of coordinates. Indeed, tensor
algebra can be viewed as linear algebra for such displacements.
Affine coordinates We will begin with an affine
coordinate system in 3D space. This simply means that the axes we use
for coordinates are three lines that intersect at the origin and that
do not lie in the same plane. These are like the usual x-y-z axes,
except that they do not need to be perpendicular. We label these axes
as x1-x2-x3. A point in space P then
has coordinates P(x1, x2, x3). If
another point Q has coordinates
Q(x1+dx1, x2+dx2,
x3+dx3),
then the displacement dx = PQ is described by the column
vector [dx1 dx2 dx3]T
relative to a basis of displacements corresponding to the
x1-x2-x3 system. In particular, the
three displacements fj corresponding to
PQj, where the Qj's are given by
Q1(x1+1, x2, x3),
Q2(x1, x2+1, x3),
Q3(x1, x2, x3+1),
provides a basis for the 3D displacements, as long as we are using an
affine system of coordinates. We let B =
{f1,f2,f3} be
the basis formed by these displacements.
Dual space The dual space to the space of 3D displacements
includes physical quantities such a work done by a force in
displacing a mass dx. Recall that the dual space is composed of linear
functions that take vectors to scalars. For the basis B =
{f1,f2,f3}, the
dual basis B* =
{f1,f2,f3}
satisfies fj(fj) = 1 and
fk(fj) = 0 for j not equal to
k. Let L be a linear functional on 3D displacements. Expand
L using B* to get
L = y1f1 +
y2f2 + y3f3,
and thus the coordinate vector for L relative to B* is
[L]B* = [y1 y2
y3]T.
Recall that if dx is a displacement (i.e., a vector), then
L(dx) =
[L]B*T[dx]B
Now, L(dx) is independent of our choice of bases,
so if we change everything to a new basis B', then we will have
L(dx) =
[L]B'*T[dx]B'.
Since [dx]B' = J[dx]B, we see that
[L]B*T[dx]B =
[L]B'*TJ[dx]B =
(JT[L]B'*)T
[dx]B,
which holds for all displacements dx. Hence,
([L]B* -
JT[L]B'*)T[dx]B
= 0 holds for all [dx]B. By choosing appropriate
values for [dx]B, we obtain
[L]B* = JT[L]B'*.
Finally, we arrive at the transformation law for
[L]B*:
[L]B'* =
(JT)-1[L]B*.
This also provides us with the transformation law for the dual basis
itself.
fj' = Jj,1f1 +
Jj,2f2 +
Jj,3f3, j=1,2,3.
Summary In the table below, we list the transformation laws
for bases and components. In all cases, the coefficients refer to
equating primed quantities to linear combinations of
unprimed quantities -- e.g., [dx]B' =
J[dx]B.
Displacements
Dual space
Basis
(JT)-1
J
Components
J
(JT)-1
Looking at the table above, we see that that there are two types of
transformation laws. Quantities that transform the same way as the
basis vectors B are called covariant. The components of
displacements transform in a way roughly inverse to the basis. For
this reason, they are called contravariant. The vectors in
the dual basis are contravariant, and the components of dual vectors
transform covariantly. Covariant quantities are denoted by subscripts,
and contravariant quantities by superscripts. Thus
fj is covariant and fj is
contravariant.
The metric tensor
Distance The distance between two points is space is the
square root of
ds2 = dx·dx,
where the "dot" denotes the usual dot product in 3D. We want to put
this in terms of components. It is in fact a little easier to look at
the dot product of two displacements,
dx·dy, where
dx =
dx1f1 +
dx2f2 +
dx3f3
and
dy =
dy1f1 +
dy2f2 +
dy3f3
Taking the dot product of these vectors results in a quadratic form
with nine terms,
dx· dy =
f1·f1
dx1dy1 +
f2·f2
dx2dy2 +
f3·f3
dx3dy3 +
f1·f2
dx1dy2 +
f2·f1
dx2dy1 +
f1·f3
dx1dy3 +
f3·f1
dx3dy1 +
f2·f3
dx2dy3 +
f3·f2
dx3dy2.
We want to put this in matrix form. Define the matrix g with entries
gj,k =
fj·fk.
The result is that we can write dx·dy as
dx·dy =
[dy]BTg[dx]B
Transformation
laws The matrix g contains the components of the metric
tensor relative to the basis B. Using the invariance of the dot
product or the transformation laws for basis vectors,one can easily
show that
g'=(JT)-1g J-1.
Equivalently, we can write out the transformation laws for the
components:
g'j,k = SUMm,ngm,n
(JT)-1m,j
(JT)-1n,k
Simple properties of the metric tensor g We close by noting
that the matrix g is actually a Gram matrix. As such is
invertible and symmetric. In addition, it has the property that it is
positive definite. All of its eigenvalues are positive.
Reciprocal basis Let B =
{f1,f2,f3} be a
basis for the displacements in 3D. We want to define a new basis for
the displacements, one that behaves like a dual basis but that is
still in 3D. We are looking for vectors
{f1,f2,f3}
with the property that
fj·fj = 1
and
fj·fk = 0, if j is
not equal to k. Every basis can be written as linear combinations of
vectors in B. That is, for any other basis, there will be coefficients
aj,k such that for j=1,2,3,
fj = aj,1f1 +
aj,2f2 +
aj,3f3
In particular, we have
fj·fk =
aj,1f1·fk
+
aj,2f2·fk
+
aj,3f3·fk
Now, recall that the metric tensor gj,k =
fj·fk. Hence, we may
rewrite the set of equations above as
fj·fk =
aj,1g1,k + aj,2g2,k +
a3,3g3,k
The sum above is just the j,k component of the matrix product ag. The
conditions for a reciprocal basis will be satisfied if and only if
[ag]jj = 1 and [ag]jk = 0
for j not equal to k. This means that ag = I, the identity
matrix. Hence, the new basis will be reciprocal if and only if a =
g-1. The matrix g-1 gives rise to a new tensor
that is called the conjugate of the metric tensor g. The
components of g-1 will be denoted by
gj,k .
The reciprocal basis it generates is then written as
fj = gj,1f1 +
gj,2f2 +
gj,3f3
Connection with dual space Let L be a linear
functional on the space of displacements. Recall that we can write
L in terms of the dual basis; its components are then
Lk. The result of applying L to dx is
L(dx) =
[L]B*T[dx]B =
L1 dx1 +
L2 dx2 +
L3 dx3
This is precisely the same result we would get from
L·dx if we regarded L as a
displacement, with reciprocal basis representation
L = L1f1 +
L2f2 +
L3f3.
The point is that linear functionals in the dual space can thus be
identified with displacements. In addition, the dual basis may be
identified with the reciprocal basis. This also means that if the
underlying basis B is changed to B', then the dual basis B* and the
reciprocal basis Brecip transform in exactly the same way
to B'* and B'recip.
Contravariant and covariant components of a vector
Two ways of representing vectors We can represent a
displacement vector v in two ways. First, we can use the basis
B = =
{f1,f2,f3}. This
gives us the following expression:
v = v1f1 +
v2f2 + v3f3,
and so [v]B = [v1 v2
v3]T
The second way is to use the reciprocal basis Brecip =
{f1,f2,f3},
which results in
v = v1f1 +
v2f2 + v3f3,
and so [v]Brecip = [v1
v2 v3]T.
The relationship between the components is determined by the
change-of-basis matrix that relates B and Brecip. As we
have seen above, the reciprocal basis is expressed in terms of B using
the entries of g-1. The formulas derived earlied for making
a change of basis apply here,
too. We identify B here with the B used earlier, and Brecip
with the basis D. Thus we identitfy g-1 and the matrix
CT used earlier, and we obtain [v]B =
(g-1)T[v]Brecip.
However, both g and g-1 are symmetric, so that in fact we
have
[v]B =
g-1[v]Brecip or, equivalently,
[v]Brecip = g[v]B
Transformation laws If we introduce a coordinate
transformation that introduces a new basis B', then we have seen that
the metric tensor transforms according to the law
g'=(JT)-1g J-1 and that
[v]B' = J[v]B. It follows that
[v]B'recip = g'[v]B' =
(JT)-1g J-1J[v]B =
(JT)-1g[v]B =
(JT)-1[v]Brecip.
Thus the components relative to the reciprocal basis transform
covariantly.
Basis vectors Let qj =
qj(x1, x2, x3), j=1,2,3,
be a set of generalized coordinates. The xj's are cartesian
coordinates. There are three coordinate curves associated with the
qj's. If x is the usual radius vector to a point in
three dimensional space, then x = x(q1,
q2, q3), and the three curves are
x = x(t, q2, q3)
(q1-curve)
x = x(q1, t, q3)
(q2-curve)
x = x(q1, q2, t)
(q3-curve)
Let's look at the q1-curve. The velocity vector tangent to
this curve at any time t is dx/dt. This can be computed using
the chain rule:
Set t = q1. This gives us our first basis vector,
e1. The others are defined in the same way. That is,
for j=1,2,3, we set
Together, these form a basis for the 3D displacements. The basis
vectors, however, do depend on the point described by the
coordinates q1, q2, q3. This means
that a different basis is associated with each point in three
dimensional space.
Reciprocal basis vectors The reciprocal basis for
{e1, e2, e3} can
be obtained via the cross product. For example,
e1 =
W-1e2×e3,
W =
e1·e2×e3.
The others are defined by cyclic permutation of the indices
involved. (W stays the same, of course. See problem 4 in Assignment 4.)
There is another way to view the reciprocal basis vectors, a way that
is similar to viewing basis vectors as tangent vectors to the
coordinate curves. Let F(x1, x2, x3)
= C be the level surfaces for for a function F. Recall that at a point
P(x1, x2, x3), the vector ∇F is
normal to the plane tangent to F=C at P. Applying this to the
coordinate surface q3(x1, x2,
x3) = c3 = constant, we see that
∇q3 is perpendicular to the tangent vectors to the
q1 and q2 coordinate curves. Since these tangent
vectors are precisely the two basis vectors e1 and
e2, it follows that ∇q3 is parallel
to e1×e2 and hence to
e3. In fact, we will show that they are equal.
Proposition For j=1,2,3,
∇qj = ej
Proof: We will show the case j=3. The others are
identical. The q3 coordinate curve through a point P is
given by x = x(q1, q2, t). On this
curve, we have q3(x) = t.
If we differentiate both sides with respect to t and use the chain
rule, we will get
∇q3·dx/dt = 1.
Since e3 = dx/dt in this case, we have that
∇q3·e3 = 1. As we
have already mentioned,
∇q3·e1 = 0 and
∇q3·e2 = 0. By
definition, ∇q3 = e3. This
completes the proof.
The metric tensor and volume element We remark that the
metric tensor is obtained from the dot product of two displacements
dx·dy, relative to the basis B =
{e1, e2,
e3}. The result is dx·dy
= [dy]BTg [dx]B,
where the metric tensor g has components gj,k =
ej·ek. From problem
5c, Assignment 9, we have that the
volume element is given by
dV = G½dq1dq2dq3,
where G = det(g).
The basis B Points in 3D are described by coordinates
(q1,q2,q3). The radius vector
x = x(q1,q2,q3). The
three vectors derived from the radius vector via ej
= ∂x/∂qj form a basis B =
{e1, e2,
e3}. This basis may vary from point to point in
space. Here we are concerned with only what happens at a single point.
The metric tensor Recall that ds2 =
[dx]BTg[dx]B =
∑gj,kdqjdqk, where
gj,k =
ej·ek. The inverse
of this matrix, g-1, is also important. We will denote its
enrties by gj,k. We point out that g is also the Gram
matrix for the basis B.
In the 2D example, g =
1+4(q1)2
1-2q1
1-2q1
2
and g-1 = (1+2q1)-2×
2
2q1-1
2q1-1
1+4(q1)2
The reciprocal basis Br The reciprocal basis
Br = {e1, e2,
e3} may be constructed in three different ways.
Gradientej = ∇
qj.
Cross producte1 =
W-1e2×e3,
W = e1·e2×e3, where the others are defined by
cyclic permutation of the indices involved.
Metric tensor We may directly use the definition
ej·ek =
δjk to find the reciprocal vectors in
terms of the basis B. The result is simply that
ej = ∑ gj,mem.
This is easy to verify:
ej·ek = ∑
gj,mem·ek
= ∑ gj,m gm,k =
[g-1g] jk =
δjk
Returning to our 2D example. Far and away the easiest method is
the third one above. The reciprocal basis for this case is
e1 =
2(1+2q1)-2e1 +
(2q1-1)(1+2q1)-2e2 e2 =
(2q1-1)(1+2q1)-2e1
+ (1+4(q1)2)(1 +
2q1)-2e2
These can also be written in terms of {i,j}.
Tensors & components We regard tensors as linear
transformations on the spaces associated with 3D
displacements. Because most problems involve only linear
transformations that take vectors to vectors, we will concentrate on
these. To that end, suppose that T is a linear transformation
that takes 3D displacements to 3D displacements; that is,
T(v) = w
Recall that we can represent T by a matrix, given bases for the
inputs and the outputs. We list these below.
Input basis
Output basis
Matrix
Column k
(j,k)-entry
Tensor
type
B
B
M
[T(ek)]B
Tj k
Mixed
B
Br
N
[T(ek)]Br
Tj k
Covariant
Br
Br
P
[T(ek)]Br
Tj k
Mixed
Br
B
Q
[T(ek)]B
Tj k
Contravariant
The names under the heading matrix are arbitrary labels. They
are used only here and nowhere else. Using the change of basis
formulas from the previous section, we can write all of the matrices in
terms of M, g, and g-1.
N = gM and Tj k = ∑gj
mTm k
P = gMg-1 and Tj k = ∑gj
m gk n Tm n
Q = Mg-1 and Tj k =
∑gk m Tj m
The point is that once one set of components is determined, so are all
of the rest.
Higher order tensors The order or rank of a tensor is the
number of indices reuired to specify its components. The tensors
described above are all order 2 tensors. Vectors are order 1 tensors,
and scalars are order 0 tensors. An order 4 tensor that arises in
elasticity theory relates the stress and strain (deformation) tensors
via a generalized Hooke's law. (See pg. 209, J. L. Synge and
A. Schild, Tensor Calculus, Dover, New York, 1978.) The
purely contravariant form of such a tensor would have four superscript
indices,
Tijkl.
To change to a mixed form where the second index is covariant but the
others are contravariant, one need only multiply by gm k
and sum over k. This results in lowering the third index.
Tijml = ∑k
gm k Tijkl
To move to a completely covariant form, one uses the same operation
on all indices,
Tmnpq =
∑ijkl gm i gn j gp k
gq l Tijkl
The process may be reversed. So if we start with the third index being
covariant and the remaining contravariant, we can raise the
index as follows:
Tijkl = ∑m
gk mTijml.
The bases B′ and B′r The setting
here is that we start with underlying coordinates (q1,
q2, q3) and make a change to the set
(q'1, q'2, q'3). The key in all of
this is to first find out how the coordinate vectors for B and B' are
related. This is derived from the Jacobian matrix of the
transformation of coordinates. Namely, we have
dq′ j = ∑(∂q′
j/∂qk) dqk.
In terms of matrices, this equation means that displacements transform
this way:
[dx]B′ = ∂q′/∂q
[dx]B.
Note that by reversing the roles of q and q′, we also have
[dx]B = ∂q/∂q′
[dx]B′.
Of course, the matrices ∂q′/∂q and
∂q/∂q′ are actually inverses of each other. We have
derived the relationships among various
components of vectors earlier. In tensor notation, these are
v′j = ∑∂q k/∂q′
j vk (equivalently,
[v]B′r =
(∂q/∂q′)T
[v]Br e′j = ∑∂q
k/∂q′ jek e′ j = ∑∂q′
j/∂qkek
Higher order tensors Consider a rank 4 tensor
Tijml. Using this tensor,
we will illustrate how the components change if we change coordinates
to q′ j. The result is
T′ ijkl = ∑
(∂q′ i/∂qm) (∂q′
j/∂qn) (∂q p/∂q′
k) (∂q′ l/∂qr)
T mnpr
Other cases are treated analogously.
The differential of a scalar quantity Temperature is a good
example of a scalar quantity. To avoid notational problems, we will
use τ to designate it. We certainly know that the temperature can
vary from point to point in a region. We want to measure how much it
changes if we move from a given point x to a nearby point
x+dx. Recall from several variable calculus that
τ(x+dx) = τ(x) +
∇τ(x)·dx +
o(|dx|)
where o(|dx|) represents terms that vanish faster than
|dx|. The linear term
dτ = ∇τ·dx
is called the differential of τ at x, and
∇&tau = ∇&tau(x) is the gradient of τ.
The gradient in generalized coordinates Let us introduce
generalized coordinates, x =
x(q1,q2,q3). Thus we may
regard τ as a function of the q's. Again appealing to several
variable calculus, we have
dτ = ∑j (∂τ/∂qj)
dqj
= ∑j∑k
(∂τ/∂qj) dqk
δjk
= ∑j∑k
(∂τ/∂qj) dqkej·ek
=
(∑j(∂τ/∂qj)ej)
· (∑kekdqk)
=
(∑j(∂τ/∂qj)ej)
·dx
From this, we see that the gradient has the following expression in
generalized coordinates:
∇τ =
∑j(∂τ/∂qj)ej.
For example, in cylindrical coordinates, we would have
∇τ = ∂τ/∂r er +
∂τ/∂θ eθ
+ ∂τ/∂z ez.
Curves and Green's Theorem To state Green's theorem,
we need to discuss simple, closed curves. These are closed curves,
like circles, but the do not intersect themselves. Rectangles,
triangles, circles, and ellipses are simple closed curves; figure
eights are not. Simple closed curves divide the plane into two
nonoverlapping regions, one interior and the other exterior. It forms
the boundary of both regions. We will consider simple closed curves
that are piecewise smooth, which just means that we are allowing a
finite number of corners. We also say that a simple closed curve is
positively oriented if it is travered in the counterclockwise
direction. Here is the statement of Green's Theorem:
Green's Theorem Let C be a piecewise smooth simple closed curve
that is the boundary of its interior region R. If F(x,y) =
A(x,y)i + B(x,y)j is a vector-valued function that is
continuously differentiable on and in C, then
Surface integrals
Surfaces See my notes, Surfaces. In addition
to discussing ways of representing surfaces, we discussed computing
surface area elements, normals to surfaces, and related topics.
Flux integrals Consider the steady state velocity
field V(x) of a fluid. We want to calculate the amount
of fluid crossing a surface parametrized by x =
x(u1, u2). Let f1 and
f2 be partials of x with respect to the
parameters u1and u2. We consider an element of
surface area, shown below as the base of the parallelepiped. Our first
step is to calculate the fluid crossing this surface element. In time
t to t+dt, the volume of fluid crossing the base of the parallelepiped
equals its volume,
(Vdt)·f1×f2
du1du2.
Vdt f1du1f2du2
The mass of the fluid crossing the base in time t to dt is then
density×volume, or
(µVdt)·f1×f2
du1du2
Thus the mass per unit time crossing the base is
F·N du1du2, where F
= µV, and N =
f1×f2 is the standard
normal. Recall that the area of the surface element is dS =
|N|du1du2. Consequently the mass per unit
time crossing the base is F·n dS, where n
is the unit normal. Integrating over the whole surface then yields
The curl and Stokes' Theorem Let S be a surface in 3D
bounded by a simple closed curve C. We will not be absolutely precise
here. One should think of S as a butterfly net, with C as its
rim. Such a surface is orientable, and we always have a
consistent piecewise continuous unit normal n defined on S. We
say that C is positively oriented if in traversing C with the
surface on our left, we are standing in the direction of n.
To state this theorem, we also need to define the curl
of a vector field
F(x)=A(x,y,z)i + B(x,y,z)j
+C(x,y,z)k.
We will assume that F has continuous partial derivatives. The
curl is then defined by
There is a useful physical interpretation for the curl. Suppose that a
fluid is rotating about a fixed axis with angular velocity
ω. Define ω to be the vector with magnitude ω
and with direction along the axis of rotation. The velocity of an
element of the fluid located at the position with radius vector
x is v(x) = ω×x. With a
little work, one can show that ω =
½∇×v. Thus one half of the curl of the
velocity vector v is the vector ω mentioned above.
Stokes' Theorem Let S be an orientable surface bounded by
a simple closed positively oriented curve C. If F is a
continuously differentiable vector-valued function defined in a region
containing S, then
Example Verify Stokes's Theorem for the vector field
F(x) = 2yi + 3xj - z2k
over the surface s, where S is the upper half of a sphere
x2+y2+z2 = 9 and C is its boundary in
the xy-plane, the circle x2+y2 = 9. C is
traversed counterclockwise.
We will first compute the line integral over C. In the xy-plane, C is
parameterized via
x(t) = 3 cos(t) i + 3 sin(t) j, 0 ≤ t
≤ 2π,
and so we have:
We now turn to finding the surface integral. ∫∫S
∇×F·n dσ. The normal compatible
with the orientation of C is n = x/|x| =
x/3. Thus, on the surface of the hemisphere S, we have
n = x(θ,φ)/3 = (3
sin(θ)cos(φ) i + 3 sin(θ)sin(φ) j +
3 cos(θ) k)/3,
and hence,
n = sin(θ)cos(φ) i + sin(θ)cos(φ)
j + cos(θ) k)
where 0 ≤ θ ≤ ½π and 0 ≤ φ ≤ 2π. Also,
the area element is dσ =
32sin(θ)dφdθ. Moreover, it is easy to show
that &nabla×F = k. We are now ready to do the
surface integral involved:
The divergence of a vector field and the Divergence
Theorem The divergence of a vector field
F(x)=A(x,y,z)i + B(x,y,z)j
+C(x,y,z)k is defined by
Like the curl, the divergence of F has a physical
interpretation in terms of fluids. This will be made clearer
later. Here is the statement of the Divergence Theorem.
To do this we must compute both integrals in the Divergence
Theorem. We will first do the volume integral. It is easy to check
that ∇·F = 3+1+2=6. Hence, we have that
∫∫∫V∇·FdV =
∫∫∫V6dV = 6π·42·5 =
480π
The surface integral must be broken into three parts: one for the top
cap, a second for the curved sides, and a third for the bottom cap.
∫∫SF·ndσ =
∫∫topF·ndσ +
∫∫sidesF·ndσ +
∫∫bottomF·ndσ
The outward normals for the top and bottom caps are k and
−k, respectively. For the top (z = 5), we are integrating
F(x,y,5)·k = 2·5 = 10, and for the
bottom (z = 0), F(x,y,0)·(−k) =
−2·0 = 0. Hence, we have
∫∫topF·ndσ =
∫∫top 10dσ = 10π42 = 160π
∫∫bottomF·ndσ =
∫∫bottom 0dσ = 0
The integral over the curved sides will require a little more
effort. The outward normal (see my notes, Surfaces,
pg. 5) and area element are, respectively,
n = cos(θ)i + sin(θ)j and dσ =
4dθdz.
In addition, on the curved sides
F(4cos(θ),4sin(θ),z) = 12cos(θ)i +
sin(θ)j + 2zk, so F·n
= 12cos2(θ) + 4sin2(θ).
The surface integral over the curved sides is then given by
∫∫sidesF·ndσ =
∫05∫02π
(12cos2(θ) + 4sin2(θ))4dθdz =
5·4(12π+4π)= 320π.
Combining these three integrals, we obtain
∫∫SF·ndσ =
160π+320π + 0 = 480π,
which agrees with the result from the volume integral. Thus we have
verified the Divergence Theorem in this case.
Equation of continuity for fluids Suppose that in a
region a fluid has a velocity field v(x,t) and density
ρ(x,t), and that there are no sources or sinks in the
region. Recall that last class we
showed that the amount of fluid crossing a surface in the direction
n per unit time is the flux,
Φ =
∫∫Sρv·ndσ.
If S is a closed surface forming the boundary of a volume V, then
Φ is the negative of the total rate of change of mass in the fluid
in V. Thus, we have the equation
∫∫Sρv·ndσ =
− d/dt ∫∫∫VρdV = −
∫∫∫V∂ρ/∂t dV.
If we use the divergence theorem to replace the surface integral by a
volume integral, we obtain
∫∫∫V ∇·(ρv)dV =
− ∫∫∫V∂ρ/∂t dV,
and, consequently, that
∫∫∫V
(∇·(ρv)+∂ρ/∂t) dV = 0
holds for every choice of V within the region under consideration. If
we take V to be a small sphere of radius ε and center
x, then the limit as ε tends to 0 of
V-1∫∫∫V
(∇·(ρv)+∂ρ/∂t) dV
is ∇·(ρv)+∂ρ/∂t. On the
other hand, this limit is of course 0. Therefore,
∇·(ρv)+∂ρ/∂t = 0.
This partial differential equation is called the equation of
continuity.
Derivation of the heat equation See § VI.2 in
Zachmanoglou and Thoe (Z/T). The steady state version of this equation
is Laplace's equation. The heat equation comes from considering energy
balance. To specify a temperature in a body, we must also take into
account two other factors: the past history of the body and the
interaction of the body with the environment. For materials without
"memory," the past history is adequately described via specifying the
temperature throughout the body at some initial time, say t = 0. The
interaction of the the environment with the body is modeled through
the use of boundary conditions.
Types of boundary conditions There are three common
types of boundary conditions.
Dirichlet boundary conditions. These specify the temperature on
the surface of the body. For example, putting an object in ice keeps
the temperature at its surface at 0 degrees C.
Neumann boundary conditions. These specify the flow of heat
across the boundary using Fourier's law. That is, they specify
n·∇u on the surface of the body. An
insulated boundary would have no heat flow, and one would require
n·∇u = 0.
Robin boundary conditions. These specify a linear combination of
u and n·∇u on the surface. They come from
Newton's law of cooling, for example.
General second order linear PDEs In addition to the heat
equation and Laplace's equation, which we derived earlier, there is a
third important type of PDE, the wave equation.
c-2∂2u/∂t2 =
∇2u.
These three equations are special cases of three general types of
second order linear PDEs. The most general form of a second order
linear PDE is
∑ajk∂2u/∂xj∂xk
+ ∑bj∂u/∂xj + cu +d = 0.
The classification scheme is based on the signs of the eigenvalues of
the symmetric matrix A with entries ajk. In the table
below, we classify PDEs for three space variables (x= x1, y
= x2, z = x3) and one time variable (t =
x4).
Classification of PDEs
Type
Eigenvalues of A
Example
Variables
Parabolic
+++0
Heat equation
3 space, 1 time
Elliptic
+++
Laplace's equation
3 space
Hyperbolic
+++-
Wave equation
3 space, 1 time
We remark that if the general PDE is multiplied by a minus sign, the
patterns in the table will have "+" replaced by"−". In general,
the solutions to the various types of equations behave like the
corresponding example. For instance, hyperbolic equations have
solutions that propagate in time, like those for the wave equation,
while parabolic equations have solutions that behave like temperature
in a heat flow problem. For further discussion, see section V.8 in Z/T.
Separation of variables
Laplace's equation We
want to solve for the steady state temperature u in a disk of radius r
= a, given Dirichlet boundary conditions. (See also section V.7 in
Z/T). We will use polar coordinates. The precise problem is this:
Because we are using polar coordinates, which are singular at r = 0
and have a discontinuity at θ = ±π, we have two
additional "boundary conditions" -- namely that u(r,θ) is well
behaved (bounded) as r approaches 0 and that u is 2π periodic in
θ.
If we ignore the nonhomogeneous boundary condition,
u(a,θ) = f(θ), then the set of solutions is a vector
space. Our aim is to construct a basis for this space. Separation
of variables is a method for finding a basis. Once we have
accomplished this, we then find the linear combination that also
satisfies the nonhomogeneous condition.
Separating variables We begin by looking for
special solutions to the homogeneous problem,
The solutions that we want have the form u(r,θ) =
R(r)Θ(θ). Plugging into the equation gives us
r-1[rR′]′Θ +
r-2RΘ″ = 0
If we now multiply this equation by r2 and divide by
RΘ, we arrive at this equation:
r[rR′]′/R +
Θ″/Θ = 0
Since r[rR′]′/R is a function of r only, and since
Θ″/Θ is a function of θ only, it follows that
both are constant. If we let μ = r[rR′]′/R, then
Θ″/Θ = -μ. With a little algebra, we obtain the
separation equations,
r[rR′]′ - μR = 0 and Θ″ + μΘ = 0.
The eigenvalue problem We now turn to the two
remaining conditions. The first of these will be satisfied if R(r) is
chosen so as to be continuous at r = 0. We will deal with it
later. The condition that u(r,θ) be 2π periodic implies that
Θ(θ) is 2π periodic. This imposes restrictions on the
possible values of μ and gives us the following eigenvalue problem:
Find all possible values of μ for which the problem
Θ″ + μΘ = 0 and Θ(θ) =
Θ(θ+2π)
has a nonzero solution Θ. These values of μ are called
eigenvalues, while the corresponding solutions Θ are
called eigenfunctions.
We can immediately eliminate μ < 0. The solutions to
Θ″ − |μ|Θ = 0 are linear combinations of
exp(±|μ|½θ), which always will blow
up as θ approaches either +∞ or −∞ or
both. They therefore cannot be periodic. For μ = 0, we do have a
single periodic solution, namely Θ = 1. The second solution is
Θ(θ) = θ, which is not periodic.
This leaves the case in which μ > 0. The differential equation
Θ″ + μΘ = 0 has two solutions,
sin(μ½θ) and
cos(μ½θ). These solutions are periodic with
fundamental period 2πμ−½. They will also
have 2π as a period if and only if some integer multiple
of 2πμ−½ is 2π. Thus, we μ > 0
is an eigenvalue if and only if there is an integer n > 0 such that
2πμ−½n = 2π. It follows that μ =
n2 and that Θ(θ) is a linear combination of
sin(nθ) and cos(nθ).
Solution to the Eigenvalue Problem
Eigenvalues μ
Eigenfunctions &Theta(θ)
02
1
12
cos(θ), sin(θ)
22
cos(2θ), sin(2θ)
⋮
⋮
n2
cos(nθ), sin(nθ)
⋮
⋮
Separation solutions We still have to find the radial
solutions corresponding the eigenvalues we found previously. For μ
= 0, the radial equation is r[rR′]′ = 0. dividing by r, we
get [rR′]′ = 0, so rR′ = C = constant, and R′
= C/r. Integrating this gives R(r) = Cln(r) + D, where D is another
constant. The only solution that behaves nicely at r = 0 is R(r) =
constant; that is, any multiple of R(r) = 1.
When μ = n2, n ≥ 1, R satisfies the equation
r[rR′]′ - n2R = 0. Working out the derivatives,
we see that this is the equation
r2R″ + rR′ - n2R = 0,
which is a Cauchy-Euler equation. The technique for solving
it is to use assume a solution of the form R = rα and
determine α. Carrying this out, we obtain
Dividing the last equation by rα, we see that
α2 − n2 = 0, and so α =
±n and the possible solutions are linear combinations of
rn and r−n. Of these two, only
rn is bounded as r approaches 0. Thus, only R(r) =
rn can be used. The separation solutions that we have
obtained are listed in the table below.
Separation Solutions
n
R(r)
Θ(θ)
u =
Rθ
0
1
1
1
1
r
cos(θ), sin(θ)
r cos(θ), r sin(θ)
2
r2
cos(2θ), sin(2θ)
r2cos(2θ), r2sin(2θ)
⋮
⋮
⋮
⋮
n
rn
cos(nθ), sin(nθ)
rncos(nθ), rnsin(nθ)
⋮
⋮
⋮
⋮
Matching the nonhomogeneous conditions We may think
of the separation solutions as forming a basis for the solution
space. The general solution is thus
u(r,θ) = A0 +
∑n≥1(Anrncos(nθ) +
Bnrnsin(nθ)).
To match the boundary condition u(a,θ) = f(θ), we need to
find coefficients such that
f(θ) = A0 +
∑n≥1(Anancos(nθ) +
Bnansin(nθ))
holds. We have already seen that we can represent f this way via its
Fourier series. Indeed, this type of
problem was Fourier's motivation for introducing such series! All we
need to do now is to identify the Fourier coefficients for f with the
coefficients above: an = Anan and
bn = Bnan. The final solution is then
u(r,θ) = a0 +
∑n≥1(r/a)n(ancos(nθ) +
bnsin(nθ)),
where an and bn are the Fourier coefficients for
f.
An example Recall that we have calculated the Fourier
series for the periodic function f(θ), which is defined by
f(θ) = |θ| for −π ≤ θ ≤ π. The
series we found was
f(θ) = ½π - (4/π)∑k≥1
(2k−1)−2cos((2k−1)θ)
By what we said above, the temperature u(r,θ) corresponding to
this f is
u(r,θ) = ½π - (4/π)∑k≥1
(r/a)2k−1 (2k−1)−2
cos((2k−1)θ)
Separation of variables in a vibrating string problem
We also discussed separation of variables for the problem of a
vibrating string with ends clamped. This is covered in section VIII.8 of
Z/T.
| 677.169 | 1 |
Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Five hundred eight Linear Algebra textbooks are available for sale on ValoreBooks.com, fourteen used from the cheapest price of $86.52, or buy new starting at $179.44.[read more
| 677.169 | 1 |
Mathematics - Algebra (529The present work contains a full and complete treatment of the topics usually included in an Elementary Algebra. The author has endeavored to prepare a course sufficiently advanced for the best High Schools and Academies, and at the same time adapted to the requirements of those who are preparing for admission to college.<br><br>Particular attention has been given to the selection of examples and problems, a sufficient number of which have been given to afford ample practice in the ordinary processes of Algebra, especially in such as are most likely to be met with in the higher branches of mathematics. Problems of a character too difficult for the average student have been purposely excluded, and great care has been taken to obtain accuracy in the answers.<br><br>The author acknowledges his obligations to the elementary text-books of Todhunter and Hamblin Smith, from which much material and many of the examples and problems have been derived. He also desires to express his thanks for the assistance which he has received from experienced teachers, in the way of suggestions of practical value.
Florian Cajori's A History of Mathematics is a seminal work in American mathematics. The book is a summary of the study of mathematics from antiquity through World War I, exploring the evolution of advanced mathematics. As the first history of mathematics published in the United States, it has an important place in the libraries of scholars and universities. A History of Mathematics is a history of mathematics, mathematicians, equations and theories; it is not a textbook, and the early chapters do not demand a thorough understanding of mathematical concepts. The book starts with the use of mathematics in antiquity, including contributions by the Babylonians, Egyptians, Greeks and Romans. The sections on the Greek schools of thought are very readable for anyone who wants to know more about Greek arithmetic and geometry. Cajori explains the advances by Indians and Arabs during the Middle Ages, explaining how those regions were the custodians of mathematics while Europe was in the intellectual dark ages. Many interesting mathematicians and their discoveries and theories are discussed, with the text becoming more technical as it moves through Modern Europe, which encompasses discussion of the Renaissance, Descartes, Newton, Euler, LaGrange and Laplace. The final section of the book covers developments in the late 19th and early 20th Centuries. Cajori describes the state of synthetic geometry, analytic geometry, algebra, analytics and applied mathematics. Readers who are not mathematicians can learn much from this book, but the advanced chapters may be easier to understand if one has background in the subject matter. Readers will want to have A History of Mathematics on their bookshelves.
Isaac Todhunter's Algebra for Beginners: With Numerous Examples is a mathematics textbook intended for the neophyte, an excellent addition to the library of math instructionals for beginners. Todhunter's textbook has been divided into 44 chapters. Early chapters highlight the most basic principles of mathematics, including sections on the principal signs, brackets, addition, subtraction, multiplication, division, and other topics that form the foundation of algebra. Simple equations make up the large majority of the material covered in this textbook. Later chapters do introduce quadratics, as well as other more advanced subjects such as arithmetical progression and scales of notation. It is important to note that Todhunter sticks very much to the basics of algebra. The content of this book lives up to its title, as this is very much mathematics for beginners. The content is provided in an easy to follow manner. This book could thus be used for independent learning as well as by a teacher. A great deal of focus has clearly been given to providing examples. Each concept is accompanied by numerous sample questions, with answers provided in the final chapter of the book. The example questions are every bit as important as the explanations, as one cannot begin to grasp mathematical concepts without having the opportunity to put them into practice. The basics of algebra are explained in an easy to follow manner, and the examples provided are clear and help to expand the knowledge of the learner. If given a chance, Isaac Todhunter's Algebra for Beginners: With Numerous Examples can be a valuable addition to your library of mathematics textbooks.
Bertrand Russell was a British logician, nobleman, historian, social critic, philosopher, and mathematician. Known as one of the founders of analytic philosophy, Russell was considered the premier logician of the 20th century and widely admired and respected for his academic work. In his lifetime, Russell published dozens of books in wildly varying fields: philosophy, politics, logic, science, religion, and psychology, among which The Principles of Mathematics was one of the first published and remains one of the more widely known. Although remembered most prominently as a philosopher, he identified as a mathematician and a logician at heart, admitting in his own biography that his love of mathematics as a child kept him going through some of his darkest moments and gave him the will to live. With his book The Principles of Mathematics, Russell aims to instill the same deep seated passion for mathematics and logic that he has carefully cultivated in the reader. He adeptly explores mathematical problems in a logical context, and attempts to prove that the study of mathematics holds critical importance to philosophy and philosophers. Russell utilizes the text to explore the some of the most fundamental concepts of mathematics, and expounds on how these building blocks can easily be applied to philosophy. In the second part of the book, Bertrand addresses mathematicians directly, discussing arithmetic and geometry principles through the lens of logic, offering yet another unique and groundbreaking interpretation of a field long before considered static. This book affords new insight and application for many basic mathematical concepts, both in roots of and application to other fields of scholarly pursuit. Russell uses his book to establish a baseline of mathematical understanding and then expands upon that baseline to establish larger and more complex ideas about the world of mathematics and its connections to other fields of personal interest. The Principles of Mathematics is a very captivating glimpse into the logic and rational of one of history's greatest thinkers. Whether you're a mathematician at heart, a logician, or someone interested in the life and thoughts of Bertrand Russell, this book is for you. With an incredible amount of information on mathematics, philosophy, and logic, this text inspires the reader to learn more and discover the ways in which these very disparate fields can interconnect and create new possibilities at their intersections.
This text is prepared to meet the needs of the student who will continue his mathematics as far as the calculus, and is written in the spirit of applied mathematics. This does not imply that algebra for the engineer is a different subject from algebra for the college man or for the secondary student who is prepared to take such a course. In fact, the topics which the engineer must emphasize, such as numerical computations, checks, graphical methods, use of tables, and the solution of specific problems, are among the most vital features of the subject for any student. But important as these topics are, they do not comprise the substance of algebra, which enables it to serve as part of the foundation for future work. Rather they furnish an atmosphere in which that foundation may be well and intelligently laid.<br><br>The concise review contained in the first chapter covers the topics which have direct bearing on the work which follows. No attempt is made to repeat all of the definitions of elementary algebra. It is assumed that the student retains a certain residue from his earlier study of the subject.<br><br>The quadratic equation is treated with unusual care and thoroughness. This is done not only for the purpose of review, but because a mastery of the theory of this equation is absolutely necessary for effective work in analytic geometry and calculus. Furthermore, a student who is well grounded in this particular is in a position to appreciate the methods and results of the theory of the general equation with a minimum of effort.<br><br>The theory of equations forms the keystone of most courses in higher algebra. The chapter on this subject is developed gradually, and yet with pointed directness, in the hope that the processes which students often perform in a perfunctory manner will take on additional life and interestThe present book forms Volume I of a two-volume series on high school algebra and embodies an especial effort to connect the elements of algebra in a clear and forcible manner with the affairs of every-day life. To this end a large variety of applied problems have been introduced and an unusual number of illustrative diagrams used in connection with both the reading matter and the exercises. At the same time, the text follows a distinctly logical line of development, thus affording that drill in accurate thought and expression which should characterize all the mathematical work of the high school. In brief, the presentation is intended to bring algebra into most intimate connection with nature while preserving its fundamental disciplinary values. This, we believe, is the demand, and properly so, of teachers and educators to-day.<br><br>Among the special features may be noted the abundance of exercises and the care with which they have been selected and graded. Problems of more than average difficulty have usually been accompanied with a hint in order that the pupil's time and energy may be reasonably conserved. At the end of the volume will be found an extensive list of supplementary exercises suitable for class use, in addition to those given in the body of the text.
The aim of the following pages is fully stated in the Introductory Chapter. Here I need only mention that some of the reflections in this book are taken from articles of mine in the Monist of 1908 and in Nature of 1909. To the Editors of these periodicals I wish to express my thanks for allowing me again to say some things which their kindness allowed me to say before. I must also thank those of my friends who have kindly read and helpfully criticised parts of this book.
The Directly-Useful Technical Series requires a few words by of introduction. Technical books of the past have arranged themselves largely under two sections: the Theoretical and the Practical. Theoretical books have been written more for the training of college students than for the supply of information to men in practice, and have been greatly filled with problems of an academic character. Practical books have often sought the other extreme, omitting the scientific basis upon which all good practice is built, whether discernible or not. The present series is intended to occupy a midway position. The information, the problems and the exercises are to be of a directly-useful character, but must at the same time be wedded to that proper amount of scientific explanation which alone will satisfy the inquiring mind. We shall thus appeal to all technical people throughout the land, either students or those in actual practiceThe present work is intended as a sequel to our Elementary Algebra for Schools. The first few chapters are devoted to a fuller discussion of Ratio, Proportion, Variation, and the Progressions, which in the former work were treated in an elementary manner; and we have here introduced theorems and examples which are unsuitable for a first course of reading.<br><br>From this point the work covers ground for the most part new to the student, and enters upon subjects of special importance: these we have endeavoured to treat minutely and thoroughly, discussing both bookwork and examples with that fulness which we have always found necessary in our experience as teachers.<br><br>It has been our aim to discuss all the essential parts as completely as possible within the limits of a single volume, but in a few of the later chapters it has been impossible to find room for more than an introductory sketch; in all such cases our object has been to map out a suitable first course of reading, referring the student to special treatises for fuller information.<br><br>In the chapter on Permutations and Combinations we are much indebted to the Rev. W. A. Whitworth for permission to make use of some of the proofs given in his Choice and Chance.
"The education of the child must accord both in mode and arrangement with the education of mankind as considered historically; or, in other words, the genesis of knowledge in the individual must follow the same course as the genesis of knowledge in the race. To M. Comte we believe society owes the enunciation of this doctrine - a doctrine which we may accept without committing ourselves to his theory of the genesis of knowledge, either in its causes or its order." If this principle, held also by Pestalozzi and Froebel, be correct, then it would seem as if the knowledge of the history of a science must be an effectual aid in teaching that science. Be this doctrine true or false, certainly the experience of many instructors establishes the importance of mathematical history in teaching. With the hope of being of some assistance to my fellow-teachers, I have prepared this book and have interlined my narrative with occasional remarks and suggestions on methods of teaching.
Upon completion of this chapter, you should be able to do the following: LDefine exponential form and logarithmic form.2. Apply laws of multiplication, division, powers, and roots for calculating logarithms.3. Determine the characteristic and mantissa of common logarithms.4. Interpolate using logarithm tables.5. Find common logarithms, antilogarithms, and natural logarithms using logarithm tables. Introduction The basic definitions and terminology associated with the study of logarithms were discussed in Mathematics, Volume 1, Navedtra 10069-D 1.Some of these basic topics are reviewed In the following paragraphs, followed by discussion of the use of logarithm tables and natural logarithms. Review Of The most important definition to remember when dealing with logarithms is that every logarithm is an exponent. For example, since 32 is equal to 9, the logarithm of 9 to the base 3 Is 2. Stating a logarithmic relationship requires that a base be stated or implied; the various exponents that designate powers of the base are logarithms to that base.
Francis William Newman was an emeritus professor of University College in London and an honorary fellow of Worcester College, Oxford. Considered quite the renaissance man, Newman's interests ranged wildly, from writings on philosophy, English reforms, Arabic, diet, grammar, political economy, Austrian Politics, Roman History, and math. He wrote at length on every subject he found of interest, and this book, Mathematical Tracts is a testament to his very successful career as a mathematician and his eloquence as an impassioned author. At its core, this book explores many of the basics theorems and principles behind geometry, aimed at the budding mathematician to encourage interest and educate. A wonderful beginners guide, but also an interesting read for anyone wanting to refresh their foundational knowledge in geometry, this book is an easy to understand and approachable guide to mathematics. After establishing the basics, this book goes in-depth on many geometrical concepts such as the treatment of ration between quantities incommensurable and primary ideas of the sphere and circle. Newman's vast knowledge of mathematics is put to excellent use in this text, expounding on mathematical concepts and explaining them with such clarity that regardless of prior mathematical knowledge, the reader is guaranteed to understand the concepts. Newman highlights a variety of shapes such as pyramids and cones in their geometric context and explains their mathematical significance. He also expands the reader's understanding of parallel straight lines and the infinite area of a plane angle, and ends the book with a plethora of tables and helpful mathematical examples intended to further clarify the core concepts of the text. Truly a one of a kind, Mathematical Tracts is the perfect book for anyone interested in mathematics. Whether you're an early learner or a seasoned professional, you will find new information that is communicated in such a passionate and compelling way that it is impossible not to be enthused and excited about the topic. An incredibly approachable book laden with mathematical concepts that are made both interesting and exciting by the overwhelming passion of the author, this book is highly recommended for all readers.
Nevertheless, it is by no means true that we are without interest in the higher, technical, mathematical field. On the contrary, we have an interest that is far more vital than the mere supplying of technical papers which can be read only by specialists. We believe that large numbers who would become active and effective in higher mathematical research are now lost to the cause simply by reason of the fact that there are no intermediate steps up which they can climb to these heights. We believe that the Monthly has a mission to perform in holding the interest of such persons by providing mathematical literature of a stimulating character that is within their range of comprehension, and by offering an appropriate medium for the publication of worthy papers which the more ambitious among them may produce.<br><br>What we have tried to do. Having in mind the principles stated above we have during 1913 supplied 325 pages of matter, exclusive of the index to Volume XX, distributed as follows: papers involving subjects of historical interest, 87 pages; papers involving general information concerning the progress of mathematics, such as meetings of associations, book reviews, notes and news, 57 pages; topics involving pedagogical considerations, especially with regard to subject matter, 37 pages; papers involving a minimum of mathematical technicalities and dealing with topics of wide interest, 56 pages; papers of a somewhat more technical character in which, however, we have tried to have the technical terms explained for the benefit of the general reader, 38 pages; problems proposed and solved and miscellaneous questions involving difficulties actually encountered by our readers, 50 pages. We have thus tried to maintain an appropriate balancing of matter so as to conserve the interests of all our readers.<br><br>What we desire to do during the coming year. During 1914 it will be our endeavor to maintain the standards already established and to improve upon the past in every way possible. In order to do this we need the cooperation and constructive criticism of all our friends. For example, a certain reader whose opinion is greatly appreciated thinks that we should have more papers on topics in applied mathematics, and he immediately backs up his opinion by sending us a contribution which will appear in the March issue. That is what we mean by cooperation. The editors have no possible interest in this undertaking which should not appeal directly to every one who is really concerned for the development of mathematics in this country. Their responsibilities and burdens are self-imposed and without emolument, save for the satisfaction which may accrue from aiding in a cause in which they heartily believe. It is their ambition to make the Monthly render genuine service to every teacher of courses in college mathematics in this country, whether in academy, high school, normal school, college, or university; to stimulate to higher endeavor every student of mathematics, whether in school or not, who may be attracted by the papers, problems, questions or discussions published in the Monthly; and to win and hold the cooperation of all who can in any department render assistance in carrying out these plansThe theory of linear associative algebras (or closed systems of hypercomplex numbers) is essentially the theory of pairs of reciprocal linear groups (52) or the theory of certain sets of matrices or bilinear forms (53). Beginning with Hamilton's discovery of quaternions seventy years ago, there has been a rapidly increasing number of papers on these various theories. The French Encyclopedia of Mathematics devotes more than a hundred pages to references and statements of results on this subject (with an additional part on ordinary complex numbers). However, the subject is rich not merely in extent, but also in depth, reaching to the very heart of modern algebra.<br><br>The purpose of this tract is to afford an elementary introduction to the general theory of linear algebras, including also non-associative algebras. It retains the character of a set of lectures delivered at the University of Chicago in the Spring Quarter of 1913. The subject is presented from the standpoint of linear algebras and makes no use either of the terminology or of theorems peculiar to the theory of bilinear forms, matrices, or groups (aside of course from 52-54, which treat in ample detail of the relations of linear algebras to those topics).<br><br>Part I relates to definitions, concrete illustrations, and important theorems capable of brief and elementary proof. A very elementary proof is given of Frobenius's theorem which shows the unique place of quaternions among algebras. The remarkable properties of Cayley's algebra of eight units are here obtained for the first time in a simple manner, without computations. Other new results and new points of view will be found in this introductory part.<br><br>In presenting in Parts II and IV the main theorems of the general theory, it was necessary to choose between the expositions by Molien, Cartan and Wedderburn (that by Frobenius being based upon bilinear forms and hence outside our plan of treatment).
The Principles of Mathematics: Vol. 1 is a terrific introduction to the fundamental concepts of mathematics. Although the book's title involves mathematics, it is not a textbook packed with equations and theorems. Instead philosopher Bertrand Russell uses mathematics to explore the structure of logic. Russell's ultimate point is that mathematics is logic and logic itself is truth. The book is substantial and covers all subjects of mathematics. It is divided into seven sections: indefinables in mathematics, number, quantity, order, infinity and continuity, space, matter and motion. Russell covers all the major developments of mathematics and the contributions of important figures to the field. His sharp mind is evident throughout The Principles of Mathematics, as he challenges established rules and teachers readers how to think through difficult problems using logic. Russell was one of the great minds of the 20th Century. In this book he discusses how his ideas were influenced by the logician Peano. He also debates other philosophers and mathematicians, and even anticipates the Theory of Relativity, which had not yet been published by Einstein. One does not need to love mathematics to gain insights from The Principles of Mathematics: Vol. 1. Those who are interested in logic, intellectualism, philosophy or history will find significant insights into logical principles. Readers who desire an intellectual challenge will truly enjoy The Principles of Mathematics: Vol. 1.
This, it may be remarked, does not arise from any analogy existing in the nature of the operations which may be totally dissimilar, but merely from the fact that they are all subject to the same laws of combination. It is true that these laws have been in many cases suggested (as Mr. Peacock has aptly termed it) by the laws of the known operations of number; but the step which is taken from arithmetical to symbolical algebra is, that leaving out of view the nature of the operations which the symbols we use represent, we suppose the existence of classes of unknown operations subject to the same laws. We are thus able to prove certain relations between the different classes of operations, which, when expressed between the symbols, are called algebraical theorems. And if we can show that any operations in any science are subject to the same laws of combination as these classes, the theorems are true of these as included in the general case; provided always that the resulting combinations are all possible in the particular operation under consideration. It will be observed that he places algebra on a formal basis; for its symbols are defined, not to represent real operations, but by laws of combination arbitrarily chosen. In a subsequent paper, however, entitled On a Difficulty in the Theory of Algebra, he practically gave up the formal view, and appears inclined to adopt the realist view instead. He says: In previous papers on the theory of algebra I have maintained the doctrine that a symbol is defined algebraically when its laws of combination are given; and that a symbol represents a given operation when the laws of combination of the latter are the same as those of the former. This, or a similar theory of the nature of algebra seems to be generally entertained by those who have turned their attention to the subject; but without in any degree leaning on it, we may say that symbols are actually subject to certain laws of combination, though we do not suppose them to be so defined; and that ksymbol representing any operation must be subject to the same laws of combination as the operation it represents. This is a departure from conventional definitions to rules founded upon the universal properties of that which is represented.
The work, the first volume of which is now offered to the public, was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1850, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.<br><br>I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.<br><br>In the preface to my former Treatise I have given a general exposition of my reasons for distinguishing arithmetical from symbolical algebra, and of my views of the just relations which their principles bear to each other, though I did not then consider it necessary to separate the exposition of one science altogether from the other.
He seems, in composing this Treatise, to have had these thee objects in view.<br><br>I.To give the general Principles and Rules of the Science, in the shortest, and, at the same time, the most clear and comprehensive manner that was possible. Agreeable to this, though every Rule is properly exemplified, yet he does not launch out into what we may call, a Tautology of examples. He rejects some applications of Algebra, that are commonly to be met with in other writers; because the number of such applications is endless: and, however useful they may be in Practice, they cannot, by the rules of good method, have place in an elementary Treatise. He has likewise omitted the Algebraical Solution of particular Geometrical problems, as requiring the knowledge of the Elements of Geometry; from which those of Algebra ought to be kept, as they really are, entirely distinct; reserving to himself to treat of the mutual relation of the two Sciences in his Third Part, and, more generally still, in the Appendix.
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Summary: Focusing on the important ideas of geometry, this book shows how to investigate two- and three-dimensional shapes with very young students. It introduces methods to describe location and position, explores simple transformations, and addresses visualization, spatial reasoning, and the building and drawing of constructions. Activities in each chapter pose questions that stimulate students to think more deeply about mathematical ideas. The CD-ROM features fourteen arti...show morecles from NCTM publications. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. ...show less
Edition/Copyright:01 Cover: Paperback Publisher:National Council of Teachers of Mathematics Published: 01/28/2001 International: NoGWBooks Vancouver, WA
An apparently unread copy in perfect condition. Dust cover is intact, with no nicks or tears. Spine has no signs of creasing. Pages are clean and are not marred by notes or folds of any kind. The book...show more is in shrinkwrapping. ...show less
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The rich and exciting field of calculus begins with the study of derivatives. This book is a practical introduction to derivatives, filled with down-to-earth explanations, detailed examples and lots of exercises (solutions included). It takes you from the basic functions all the way to advanced differentiation rules and proofs. Check out the sample for the table of contents and a taste of the action. From the author of "Mathematical Shenanigans", "Great Formulas Explained" and the "Math Shorts" series. A supplement to this book is available under the title "Exercises to Math Shorts – Derivatives". It contains an additional 28 exercises including detailed solutions.
Note: Except for the very basics of algebra, no prior knowledge is required to enjoy this book.
So you want to learn about differential equations? Excellent choice. Differential equations are not only of central importance to science, they can also be quite stimulating and fun (that's right). In the broadest sense, a differential equation is any equation that connects a function with one or more of its derivatives. What makes these kinds of equations particularly important?
Remember that a derivative expresses the rate of change of a quantity. So the differential equation basically establishes a link between the rate of change of said quantity and its current value. Such a link is very common in nature. Consider population growth. It is obvious that the rate of change will depend on the current size of the population. The more animals there are, the more births (and deaths) we can expect and hence the faster the size of the population will change.
A commonly used model for population growth is the exponential model. It is based on the assumption that the rate of change is proportional to the current size of the population. Let's put this into mathematical form. We will denote the size of the population by N (measured in number of animals) and the first derivative with respect to time by dN/dt (measured in number of animals per unit time). Note that other symbols often used for the first derivative are N' and Ṅ. We will however stick to the so-called Leibniz notation dN/dt as it will prove to be quite instructive when dealing with separable differential equations. That said, let's go back to the exponential model.
With N being the size of the population and dN/dt the corresponding rate of change, our assumption of proportionality between the two looks like this:
with r being a constant. We can interpret r as the growth rate. If r > 0, then the population will grow, if r < 0, it will shrink. This model has proven to be successful for relatively small animal populations. However, there's one big flaw: there is no limiting value. According to the model, the population would just keep on growing and growing until it consumes the entire universe. Obviously and luckily, bacteria in Petri dish don't behave this way. For a more accurate model, we need to take into account the limits of the environment.
The differential equation that forms the basis of the logistic model, called Verhulst equation in honor of the Belgian mathematician Pierre François Verhulst, does just that. Just like the differential equation for exponential growth, it relates the current size N of the population to its rate of change dN/dt, but also takes into account the finite capacity K of the environment:
Take a careful look at the equation. Even without any calculations a differential equation can tell a vivid story. Suppose for example that the population is very small. In this case N is much smaller than K, so the ratio N/K is close to zero. This means that we are back to the exponential model. Hence, the logistic model contains the exponential model as a special case. Great! The other extreme is N = K, that is, when the size of the population reaches the capacity. In this case the ratio N/K is one and the rate of change dN/dt becomes zero, which is exactly what we were expecting. No more growth at the capacity.
Definition and Equation of Motion
Now that you have seen two examples of differential equations, let's generalize the whole thing. For starters, note that we can rewrite the two equations as such:
Denoting the dependent variable with x (instead of N) and higher order derivatives with dnx/dtn (with n = 2 resulting in the second derivative, n = 3 in the third derivative, and so on), the general form of a differential equation looks like this:
Wow, that looks horrible! But don't worry. We just stated in the broadest way possible that a differential equation is any equation that connects a function x(t) with one or more its derivatives dx/dt, d2x/dt2, and so on. The above differential equation is said to have the order n. Up to now, we've only been dealing with first order differential equations.
The following equation is an example of a second order differential equation that you'll frequently come across in physics. Its solution x(t) describes the position or angle over time of an oscillating object (spring, pendulum).
with c being a constant. Second order differential equations often arise naturally from Newton's equation of motion. This law, which even the most ruthless crook will never be able to break, states that the object's mass m times the acceleration a experienced by it is equal to the applied net force F:
The force can be a function of the object's location x (spring), the velocity v = dx/dt (air resistance), the acceleration a = d2x/dt2 (Bremsstrahlung) and time t (motor):
Hence the equation of motion becomes:
A second order differential equation which leads to the object's position over time x(t) given the forces involved in shaping its motion. It might not look pretty to some (it does to me), but there's no doubt that it is extremely powerful and useful.
Equilibrium Points
To demonstrate what equilibrium points are and how to compute them, let's take the logistic model a step further. In the absence of predators, we can assume the fish in a certain lake to grow according to Verhulst's equation. The presence of fishermen obviously changes the dynamics of the population. Every time a fisherman goes out, he will remove some of the fish from the population. It is safe to assume that the success of the fisherman depends on the current size of the population: the more fish there are, the more he will be able to catch. We can set up a modified version of Verhulst's equation to describe the situation mathematically:
with a constant c > 0 that depends on the total number of fishermen, the frequency and duration of their fishing trips, the size of their nets, and so on. Solving this differential equation is quite difficult. However, what we can do with relative ease is finding equilibrium points.
Remember that dN/dt describes the rate of change of the population. Hence, by setting dN/dt = 0, we can find out if and when the population reaches a constant size. Let's do this for the above equation.
This leads to two solutions:
The first equilibrium point is quite boring. Once the population reaches zero, it will remain there. You don't need to do math to see that. However, the second equilibrium point is much more interesting. It tells us how to calculate the size of the population in the long run from the constants. We can also see that a stable population is only possible if c < r.
Note that not all equilibrium points that we find during such an analysis are actually stable (in the sense that the system will return to the equilibrium point after a small disturbance). The easiest way to find out whether an equilibrium point is stable or not is to plot the rate of change, in this case dN/dt, over the dependent variable, in this case N. If the curve goes from positive to negative values at the equilibrium point, the equilibrium is stable, otherwise it is unstable.
Who doesn't love a day at the theme park? You can go on thrilling roller‒coaster rides, enjoy elaborate shows, have a tasty lunch in between or just relax and take in the scenery. Of course there's one thing that does spoil the fun a bit: the waiting. For the most popular attractions waiting times of around one hour are not uncommon during peak times, while the ride itself may be over in no more than two or three minutes.
Let's work towards a basic model for queues in theme parks and other situations in which queues commonly arise. We will assume that the passing rate R(t), that is, the number of people passing the entrance of the attraction per unit time, is given. How many of these will enter the line? This will depend on the popularity of the attraction as well as the current size of the line. The more people are already in the line, the less likely others are to join. We'll denote the number of people in the line at time t with n(t) and use this ansatz for the rate r(t) at which people join the queue:
The constant a expresses the popularity of the attraction (more specifically, it is the percentage of passers‒by that will use the attraction if no queue is present) and the constant b is a "line repulsion" factor. The stronger visitors are put off by the presence of a line, the higher its value will be. How does the size of the line develop over time given the above function? We assume that the maximum serving rate is c people per unit time. So the rate of change for the number of people in line is (for n(t) ≥ 0):
In case the numerical evaluation returns a value n(t) < 0 (which is obviously nonsense, but a mathematical possibility given our ansatz), we will force n(t) = 0. An interesting variation, into which we will not dive much further though, is to include a time lag. Usually the expected waiting time is displayed to visitors on a screen. The visitors make their decision on joining the line based on this information. However, the displayed waiting time is not updated in real‒time. We have to expect that there's a certain delay d between actual and displayed length of the line. With this effect included, our equation becomes:
Simulation
For the numerical solution we will go back to the delay‒free version. We choose one minute as our unit of time. For the passing rate, that is, the people passing by per minute, we set:
R(t) = 0.00046 · t · (720 ‒ t)
We can interpret this function as such: at t = 0 the park opens and the passing rate is zero. It then grows to a maximum of 60 people per minute at t = 360 minutes (or 6 hours) and declines again. At t = 720 minutes (or 12 hours) the park closes and the passing rate is back to zero. We will assume the popularity of the attraction to be:
a = 0.2
So if there's no line, 20 % of all passers‒by will make use of the attraction. We set the maximum service rate to:
c = 5 people per minute
What about the "line repulsion" factor? Let's assume that if the line grows to 200 people (given the above service rate this would translate into a waiting time of 40 minutes), the willingness to join the line drops from the initial 20 % to 10 %.
→ b = 0.005
Given all these inputs and the model equation, here's how the queue develops over time:
It shows that no line will form until around t = 100 minutes into opening the park (at which point the passing rate reaches 29 people per minute). Then the queue size increases roughly linearly for the next several hours until it reaches its maximum value of n = 256 people (waiting time: 51 minutes) at t = 440 minutes. Note that the maximum value of the queue size occurs much later than the maximum value of the passing rate. After reaching a maximum, there's a sharp decrease in line length until the line ceases to be at around t = 685 minutes. Further simulations show that if you include a delay, there's no noticeable change as long as the delay is in the order of a few minutes.
The distance to our neighboring star Alpha Centauri is roughly 4.3 lightyears or 25.6 trillion km. This is an enormous distance. It would take the Space Shuttle 165,000 years to cover this distance. That's 6,600 generations of humans who'd know nothing but the darkness of space. Obviously, this is not an option. Do we have the technologies to get there within the lifespan of a person? Surprisingly, yes. The concept of antimatter propulsion might sound futuristic, but all the technologies necessary to build such a rocket exist. Today.
What exactly do you need to build a antimatter rocket? You need to produce antimatter, store antimatter (remember, if it comes in contact with regular matter it explodes, so putting it in a box is certainly not a possibility) and find a way to direct the annihilation products. Large particle accelerators such as CERN routinely produce antimatter (mostly anti-electrons and anti-protons). Penning-Traps, a sophisticated arrangement of electric and magnetic fields, can store charged antimatter. And magnetic nozzles, suitable for directing the products of proton / anti-proton annihilations, have already been used in several experiments. It's all there.
So why are we not on the way to Alpha Centauri? We should be making sweet love with green female aliens, but instead we're still banging our regular, non-green, non-alien women. What's the hold-up? It would be expensive. Let me rephrase that. The costs would be blasphemous, downright insane, Charlie Manson style. Making one gram of antimatter costs around 62.5 trillion $, it's by far the most expensive material on Earth. And you'd need tons of the stuff to get to Alpha Centauri. Bummer! And even if we'd all get a second job to pay for it, we still couldn't manufacture sufficient amounts in the near future. Currently 1.5 nanograms of antimatter are being produced every year. Even if scientists managed to increase this rate by a factor of one million, it would take 1000 years to produce one measly gram. And we need tons of it! Argh. Reality be a harsh mistress …
Yesterday I released the second part of my "Math Shorts" series. This time it's all about integrals. Integrals are among the most useful and fascinating mathematical concepts ever conceived. The ebook is a practical introduction for all those who don't want to miss out. In it you'll find down-to-earth explanations, detailed examples and interesting applications. Check out the sample (see link to product page) a taste of the action.
Important note: to enjoy the book, you need solid prior knowledge in algebra and calculus. This means in particular being able to solve all kinds of equations, finding and interpreting derivatives as well as understanding the notation associated with these topics.
Differential equations are an important and fascinating part of mathematics with numerous applications in almost all fields of science. This book is a gentle introduction to the rich world of differential equations filled with no-nonsense explanations, step-by-step calculations and application-focused examples.
Important note: to enjoy the book, you need solid prior knowledge in algebra and calculus. This means in particular being able to solve all kinds of equations, finding and interpreting derivatives, evaluating integrals as well as understanding the notation that goes along with those.
An abstraction often used in physics is motion with constant acceleration. This is a good approximation for many different situations: free fall over small distances or in low-density atmospheres, full braking in car traffic, an object sliding down an inclined plane, etc … The mathematics behind this special case is relatively simple. Assume the object that is subject to the constant acceleration a (in m/s²) initially has a velocity v(0) (in m/s). Since the velocity is the integral of the acceleration function, the object's velocity after time t (in s) is simply:
1) v(t) = v(0) + a · t
For example, if a car initially goes v(0) = 20 m/s and brakes with a constant a = -10 m/s², which is a realistic value for asphalt, its velocity after a time t is:
v(t) = 20 – 10 · t
After t = 1 second, the car's speed has decreased to v(1) = 20 – 10 · 1 = 10 m/s and after t = 2 seconds the car has come to a halt: v(2) = 20 – 10 · 2 = 0 m/s. As you can see, it's all pretty straight-forward. Note that the negative acceleration (also called deceleration) has led the velocity to decrease over time. In a similar manner, a positive acceleration will cause the speed to go up. You can read more on acceleration in this blog post.
What about the distance x (in m) the object covers? We have to integrate the velocity function to find the appropriate formula. The covered distance after time t is:
2) x(t) = v(0) · t + 0.5 · a · t²
While that looks a lot more complicated, it is really just as straight-forward. Let's go back to the car that initially has a speed of v(0) = 20 m/s and brakes with a constant a = -10 m/s². In this case the above formula becomes:
x(t) = 20 · t – 0.5 · 10 · t²
After t = 1 second, the car has traveled x(1) = 20 · 1 – 0.5 · 10 · 1² = 15 meters. By the time it comes to a halt at t = 2 seconds, it moved x(2) = 20 · 2 – 0.5 · 10 · 2² = 20 meters. Note that we don't have to use the time as a variable. There's a way to eliminate it. We could solve equation 1) for t and insert the resulting expression into equation 2). This leads to a formula connecting the velocity v and distance x.
3)
Solved for x it looks like this:
3)'
It's a very useful formula that you should keep in mind. Suppose a tram accelerates at a constant a = 1.3 m/s², which is also a realistic value, from rest (v(0) = 0 m/s). What distance does it need to go to full speed v = 10 m/s? Using equation 3)' we can easily calculate this:
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Here are a few exercises and solutions using the equations 1), 2) and 3).
1. During free fall (air resistance neglected) an object accelerates with about a = 10 m/s. Suppose the object is dropped, that is, it is initially at rest (v(0) = 0 m/s).
a) What is its speed after t = 3 seconds? b) What distance has it traveled after t = 3 seconds? c) Suppose we drop the object from a tower that is x = 20 meters tall. At what speed will it impact the ground? d) How long does the drop take?
Hint: in exercise d) solve equation 1) for t and insert the result from c)
2. During the reentry of space crafts accelerations can be as high as a = -70 m/s². Suppose the space craft initially moves with v(0) = 6000 m/s.
a) What's the speed and covered distance after t = 10 seconds? b) How long will it take the space craft to half its initial velocity? c) What distance will it travel during this time?
3. An investigator arrives at the scene of a car crash. From the skid marks he deduces that it took the car a distance x = 55 meters to come to a halt. Assume full braking (a = -10 m/s²). Was the car initially above the speed limit of 30 m/s?
A very important metric for banner advertiesment is the CTR (click through rate). It is simply the number of clicks the ad generated divided by the number of total impressions. You can also think of it as the product of the probability of a user noticing the ad and the probability of the user being interested in the ad.
CTR = clicks / impressions = p(notice) · p(interested)
The current average CTR is around 0.09 % or 9 clicks per 10,000 impressions and has been declining for the past several years. What are the reasons for this? For one, the common banner locations are familiar to web users and are thus easy to ignore. There's also the increased popularity of ad-blocking software.
The attitude of internet users is generally negative towards banner ads. This is caused by advertisers using more and more intrusive formats. These include annoying pop-ups and their even more irritating sisters, the floating ads. Adopting them is not favorable for advertisers. They harm a brand and produce very low CTRs. So hopefully, we will see an end to such nonsense soon.
As for animated ads, their success depends on the type of website and target group. For high-involvement websites that users visit to find specific information (news, weather, education), animated banners perform worse than static banners. In case of low-involvement websites that are put in place for random surfing (entertainment, lists, mini games) the situation is reversed. The target group also plays an important role. For B2C (business-to-consumer) ads animation generally works well, while for B2B (business-to-business) animation was shown to lower the CTR.
The language used in ads has also been extensively studied. One interesting result is that often it is preferable to use English language even if the ad is displayed in a country in which English is not the first language. A more obvious result is that catchy words and calls to action ("read more") increase the CTR.
As for the banner size, there is inconclusive data. Some analysis report that the CTR grows with banner size, while others conclude that banner sizes around 250×250 or 300×250 generate the highest CTRs. There is a clearer picture regarding shape: in terms of CTR, square shapes work better than thin rectangles having the same size. No significant difference was found between vertical and horizontal rectangles.
Here's another hint: my own theoretical calculations show that higher CTRs can be achieved by advertising on pages that have a low visitor loyalty. The explanation for this counter-intuitive outcome as well as a more sophisticated formula for the CTR can be found here. It is, in a nutshell, a result of the multiplication rule of statistics. The calculation also shows that on sites with a low visitor loyalty the CTR will stay constant, while on websites with a high visitor loyalty it will decrease over time.
Sources and further reading:
Study on banner advertisement type and shape effect on click-through-rate and conversion
First of all: why should a website's visitor loyalty have any effect at all on the CTR we can expect to achieve with a banner ad? What does the one have to do with the other? To understand the connection, let's take a look at an overly simplistic example. Suppose we place a banner ad on a website and get in total 3 impressions (granted, not a realistic number, but I'm only trying to make a point here). From previous campaigns we know that a visitor clicks on our ad with a probability of 0.1 = 10 % (which is also quite unrealistic).
This demonstrates that we can expect more clicks if the website's visitor loyalty is low, which might seem counter-intuitive at first. But the great thing about mathematics is that it cuts through bullshit better than the sharpest knife ever could. Math doesn't lie. Let's develop a model to show that a higher vistor loyalty translates into a lower CTR.
Suppose we got a number of I impressions on the banner ad in total. We'll denote the percentage of visitors that contributed …
… only one impression by f(1)
… two impressions by f(2)
… three impressions by f(3)
And so on. Note that this distribution f(n) must satisfy the condition ∑[n] n·f(n) = I for it all to check out. The symbol ∑[n] stands for the sum over all n.
We'll assume that the probability of a visitor clicking on the ad is q. The probability that this visitor clicks on the ad at least once during n visits is just: p(n) = 1 – (1 – q)^n (to understand why you have the know about the multiplication rule of statistics – if you're not familiar with it, my ebook "Statistical Snacks" is a good place to start).
Let's count the expected number of clicks for the I impressions. Visitors …
And so on. So the total number of clicks we can expect is: c = ∑[n] p(n)·f(n)/n·I. Since the CTR is just clicks divided by impressions, we finally get this beautiful formula:
CTR = ∑[n] p(n)·f(n)/n
The expression p(n)/n decreases as n increases. So a higher visitor loyalty (which mathematically means that f(n) has a relatively high value for n greater than one) translates into a lower CTR. One final conclusion: the formula can also tell us a bit about how the CTR develops during a campaign. If a website has no loyal visitors, the CTR will remain at a constant level, while for websites with a lot of loyal visitors, the CTR will decrease over time.
A way of expressing a quantity in relative terms is to do the ratio with respect to a reference value. This helps to put a quantity into perspective. For example, in mechanics the acceleration is often expressed in relation to the gravitational acceleration. Instead of saying the acceleration is 22 m/s² (which is hard to relate to unless you know mechanics), we can also say the acceleration is 22 / 9.81 ≈ 2.2 times the gravitational acceleration or simply 2.2 g's (which is much easier to comprehend).
The decibel (dB) is also a general way of expressing a quantity in relative terms, sort of a "logarithmic ratio". And just like the ratio, it is not a physical unit or limited to any field such as mechanics, audio, etc … You can express any quantity in decibels. For example, if we take the reference value to be the gravitational acceleration, the acceleration 22 m/s² corresponds to 3.5 dB.
To calculate the decibel value L of a quantity x relative to the reference value x(0), we can use this formula:
In acoustics the decibel is used to express the sound pressure level (SPL), measured in Pascal = Pa, using the threshold of hearing (0.00002 Pa) as reference. However, in this case a factor of twenty instead of ten is used. The change in factor is a result of inputting the squares of the pressure values rather than the linear values.
The sound coming from a stun grenade peaks at a sound pressure level of around 15,000 Pa. In decibel terms this is:
which is way past the threshold of pain that is around 63.2 Pa (130 dB). Here are some typical values to keep in mind:
Why use the decibel at all? Isn't the ratio good enough for putting a quantity into perspective? The ratio works fine as long as the quantity doesn't go over many order of magnitudes. This is the case for the speeds or accelerations that we encounter in our daily lives. But when a quantity varies significantly and spans many orders of magnitude (which is what the SPL does), the decibel is much more handy and relatable.
Another reason for using the decibel for audio signals is provided by the Weber-Fechner law. It states that a stimulus is perceived in a logarithmic rather than linear fashion. So expressing the SPL in decibels can be regarded as a first approximation to how loud a sound is perceived by a person as opposed to how loud it is from a purely physical point of view.
Note that when combining two or more sound sources, the decibel values are not simply added. Rather, if we combine two sources that are equally loud and in phase, the volume increases by 6 dB (if they are out of phase, it will be less than that). For example, when adding two sources that are at 50 dB, the resulting sound will have a volume of 56 dB (or less).
Suppose we have an audio signal which peaks at L decibels. We apply a compressor with a threshold T (with T being smaller than L, otherwise the compressor will not spring into action) and ratio r. How does this effect the maximum volume of the audio signal? Let's derive a formula for that. Remember that the compressor leaves the parts of the signal that are below the threshold unchanged and dampens the excess volume (threshold to signal level) by the ratio we set. So the dynamic range from the threshold to the peak, which is L – T, is compressed to (L – T) / r. Hence, the peak volume after compression is:
L' = T + (L – T) / r
For example, suppose our mix peaks at L = – 2 dB. We compress it using a threshold of T = – 10 dB and a ratio r = 2:1. The maximum volume after compression is:
On April 12, 2011, something extraordinary happened. A 58-year-old woman that was paralyzed from the neck down reached for a bottle of coffee, drank from a straw and put the bottle back on the table. But she didn't reach with her own hand – she controlled a robotic arm with her mind. Uneblievable? It is. But decades of research made the unbelievable possible. Watch this exceptional and moving moment in history here (click on picture for Youtube video).
The 58-year-old women (patient S3) was part of the BrainGate2 project, a collaboration of researchers at the Department of Veterans Affairs, Brown University, German Aerospace Center (DLR) and others. The scientists implanted a small chip containing 96 electrodes into her motor cortex. This part of the brain is responsible for voluntary movement. The chip measures the electrical activity of the brain and an external computer translates this pattern into the movement of a robotic arm. A brain-computer interface. And it's not science-fiction, it's science.
During the study the woman was able to grasp items during the allotted time with a 70 % success rate. Another participant (patient T2) even managed to achieve a 96 % success rate. Besides moving robotic arms, the participants also were given the task to spell out words and sentences by indicating letters via eye movement. Participant T2 spelt out this sentence: "I just imagined moving my own arm and the [robotic] arm moved where I wanted it to go".
Almost all music and recorded speech that you hear has been sent through at least one compressor at some point during the production process. If you are serious about music production, you need to get familiar with this powerful tool. This means understanding the big picture as well as getting to know each of the parameters (Threshold, Ratio, Attack, Release, Make-Up Gain) intimately.
How They Work
Throughout any song the volume level varies over time. It might hover around – 6 dB in the verse, rise to – 2 dB in the first chorus, drop to – 8 dB in the interlude, and so on. A term that is worth knowing in this context is the dynamic range. It refers to the difference in volume level from the softest to the loudest part. Some genres of music, such as orchestral music, generally have a large dynamic range, while for mainstream pop and rock a much smaller dynamic range is desired. A symphony might range from – 20 dB in the soft oboe solo to – 2 dB for the exciting final chord (dynamic range: 18 dB), whereas your common pop song will rather go from – 8 dB in the first verse to 0 dB in the last chorus (dynamic range: 8 dB).
During a recording we have some control over what dynamic range we will end up with. We can tell the musicians to take it easy in the verse and really go for it in the chorus. But of course this is not very accurate and we'd like to have full control of the dynamic range rather than just some. We'd also like to be able to to change the dynamic range later on. Compressors make this (and much more) possible.
The compressor constantly monitors the volume level. As long as the level is below a certain threshold, the compressor will not do anything. Only when the level exceeds the threshold does it become active and dampen the excess volume by a certain ratio. In short: everything below the threshold stays as it is, everything above the threshold gets compressed. Keep this in mind.
Suppose for example we set the threshold to – 10 dB and the ratio to 4:1. Before applying the compressor, our song varies from a minimum value of – 12 dB in the verse to a maximum value – 2 dB in the chorus. Let's look at the verse first. Here the volume does not exceed the threshold and thus the compressor does not spring into action. The signal will pass through unchanged. The story is different for the chorus. Its volume level is 8 dB above the threshold. The compressor takes this excess volume and dampens it according to the ratio we set. To be more specific: the compressor turns the 8 dB excess volume into a mere 8 dB / 4 = 2 dB. So the compressed song ranges from – 12 dB in the verse to -10 dB + 2 dB = – 8 dB in the chorus.
As you can see, the compressor had a significant effect on the dynamic range. Choosing appropriate values for the threshold and ratio, we are free to compress the song to any dynamic range we desire. When using a DAW (Digital Audio Workstation such as Cubase, FL Studio or Ableton Live), it is possible to see the workings of a compressor with your own eyes. The image below shows the uncompressed file (top) and the compressed file (bottom) with the threshold set to – 12 dB and the ratio to 2:1.
The soft parts are identical, while the louder parts (including the short and possibly problematic peaks) have been reduced in volume. The dynamic range clearly shrunk in the process. Note that after applying the compressor, the song's effective volume (RMS) is much lower. Since this is usually not desired, most compressors have a parameter called make-up gain. Here you can specify by how much you'd like the compressor to raise the volume of the song after the compression process is finished. This increase in volume is applied to all parts of the song, soft or loud, so there will not be another change in the dynamic range. It only makes up for the loss in loudness (hence the name).
Usage of Compressors
We already got to know one application of the compressor: controlling the dynamic range of a song. But usually this is just a first step in reaching another goal: increasing the effective volume of the song. Suppose you have a song with a dynamic range of 10 dB and you want to make it as loud as possible. So you move the volume fader until the maximum level is at 0 dB. According to the dynamic range, the minimum level will now be at – 10 dB. The effective volume will obviously be somewhere in-between the two values. For the sake of simplicity, we'll assume it to be right in the middle, at – 5 dB. But this is too soft for your taste. What to do?
You insert a compressor with a threshold of – 6 dB and a ratio of 3:1. The 4 dB range from the minimum level – 10 dB to the threshold – 6 dB is unchanged, while the 6 dB range from the threshold – 6 dB to the maximum level 0 dB is compressed to 6 dB / 3 = 2 dB. So overall the dynamic range is reduced to 4 dB + 2 dB = 6 dB. Again you move the volume fader until the maximum volume level coincides with 0 dB. However, this time the minimum volume will be higher, at – 6 dB, and the effective volume at – 3 dB (up from the – 5 dB we started with). Mission accomplished, the combination of compression and gain indeed left us with a higher average volume.
In theory, this means we can get the effective volume up to almost any value we desire by compressing a song and then making it louder. We could have the whole song close to 0 dB. This possibility has led to a "loudness war" in music production. Why not go along with that? For one, you always want to put as much emphasis as possible on the hook. This is hard to do if the intro and verse is already blaring at maximum volume. Another reason is that severely reducing the dynamic range kills the expressive elements in your song. It is not a coincidence that music which strongly relies on expressive elements (orchestral and acoustic music) usually has the highest dynamic range. It needs the wide range to go from expressing peaceful serenity to expressing destructive desperation. Read the following out loud and memorize it: the more expression it has, the less you should compress. While a techno song might work at maximum volume, a ballad sure won't.
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Background Info – SPL and Loudness
Talking about how loud something is can be surprisingly complicated. The problem is that our brain does not process sound inputs in a linear fashion. A sound wave with twice the sound pressure does not necessarily seem twice as loud to us. So when expressing how loud something is, we can either do this by using well-defined physical quantities such as the sound pressure level (which unfortunately does not reflect how loud a person perceives something to be) or by using subjective psycho-acoustic quantities such as loudness (which is hard to define and measure properly).
Sound waves are pressure and density fluctuations that propagate at a material- and temperature-dependent speed in a medium. For air at 20 °C this speed is roughly 340 m/s. The quantity sound pressure expresses the deviation of the sound wave pressure from the pressure of the surrounding air. The sound pressure level, in short: SPL, is proportional to the logarithm of the effective sound pressure. Long story short: the stronger the sound pressure, the higher the SPL. The SPL is used to objectively measure how loud something is. Another important objective quantity for this purpose is the volume. It is a measure of how much energy is contained in an audio signal and thus closely related to the SPL.
A subjective quantity that reflects how loud we perceive something to be is loudness. Due to our highly non-linear brains, the loudness of an audio signal is not simply proportional to its SPL or volume level. Rather, loudness depends in a complex way on the SPL, frequency, duration of the sound, its bandwidth, etc … In the image below you can see an approximation of the relationship between loudness, SPL and frequency.
Any red curve is a curve of equal loudness. Here's how we can read the chart. Take a look at the red curve at the very bottom. It starts at 75 dB SPL and a frequency of 20 Hz and reaches 25 dB SPL at 100 Hz. Since the red curve is a curve of equal loudness, we can conclude we perceive a 75 dB SPL sound at 20 Hz to be just as loud as a 25 dB SPL sound at 100 Hz, even though from a purely physical point of view the first sound is three times as loud as the second (75 dB / 25 dB = 3).
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(Compressor in Cubase)
Threshold and Ratio
What's the ideal threshold to use? This depends on what you are trying to accomplish. Suppose you set the threshold at a relatively high value (for example – 10 dB in a good mix). In this case the compressor will be inactive for most of the song and only kick in during the hook and short peaks. With the threshold set to a high value, you are thus "taking the top off". This would be a suitable choice if you are happy with the dynamics in general, but would like to make the mix less aggressive.
What about low thresholds (such as -25 dB in a good mix)? In this case the compressor will be active for the most part of the song and will make the entire song quite dense. This is something to consider if you aim to really push the loudness of the song. Once the mix is dense, you can go for a high effective volume. But a low threshold compression can also add warmth to a ballad, so it's not necessarily a tool restricted to usage in the loudness war.
Onto the ratio. If you set the ratio to a high value (such as 5:1 and higher), you are basically telling the mix: to the threshold and no further. Anything past the threshold will be heavily compressed, which is great if you have pushy peaks that make a mix overly aggressive. This could be the result of a snare that's way too loud or an inexperienced singer. Whatever the cause, a carefully chosen threshold and a high ratio should take care of it in a satisfying manner. Note though that in this case the compressor should be applied to the track that is causing the problem and not the entire mix.
A low value for the ratio (such as 2:1 or smaller) will have a rather subtle effect. Such values are perfect if you want to apply the compressor to a mix that already sounds well and just needs a finishing touch. The mix will become a little more dense, but its character will be kept intact.
Attack and Release
There are two important parameters we have ignored so far: the attack and release. The attack parameter allows you to specify how quickly the compressor sets in once the volume level goes past the threshold. A compressor with a long attack (20 milliseconds or more) will let short peaks pass. As long as these peaks are not over-the-top, this is not necessarily a bad thing. The presence of short peaks, also called transients, is important for a song's liveliness and natural sound. A long attack makes sure that these qualities are preserved and that the workings of the compressor are less noticeable.
A short attack (5 milliseconds or less) can produce a beautifully crisp sound that is suitable for energetic music. But it is important to note that if the attack is too short, the compressor will kill the transients and the whole mix will sound flat and bland. Even worse, a short attack can lead to clicks and a nervous "pumping effect". Be sure to watch out for those as you shorten the attack.
The release is the time for the compressor to become inactive once the volume level goes below the threshold. It is usually much longer than the attack, but the overall principles are similar. A long release (600 milliseconds or more) will make sure that the compression happens in a more subtle fashion, while a short release (150 milliseconds or less) can produce a pumping sound.
It is always a good idea to choose the release so that it fits the rhythm of your song (the same of course is true for temporal parameters in reverb and delay). One way to do this is to calculate the time per beat TPB in milliseconds from your song's tempo as measured in beats per minute BPM and use this value as the point of reference.
TPB [ms] = 60000 / BPM
For example, in a song with the tempo BPM = 120 the duration of one beat is TPB = 60000 / 120 = 500 ms. If you need a longer release, use a multiple of it (1000 ms, 1500 ms, and so on), for a shorter release divide it by any natural number (500 ms / 2 = 250 ms, 500 ms / 3 = 167 ms, and so on). This way the compressor will "breathe" in unison with your music.
If you are not sure where to start regarding attack and release, just make use of the 20/200-rule: Set the attack to 20 ms, the release to 200 ms and work towards the ideal values from there. Alternatively, you can always go through the presets of the compressor to find suitable settings.
This book is a quick guide to effects, mixing and mastering for beginners using Cubase as its platform. The first chapter highlights the most commonly used effects in audio production such as compressors, limiters, equalizers, reverb, delay, gates and others. You will learn about how they work, when to apply them, the story behind the parameters and what traps you might encounter. The chapter also contains a quick peek into automation and what it can do.
In the second chapter we focus on what constitutes a good mix and how to achieve it using a clear and comprehensible strategy. This is followed by a look at the mastering chain that will help to polish and push a mix. The guide is sprinkled with helpful tips and background information to make the learning experience more vivid. You get all of this for a fair price of $ 3.95.
For temperature there is a definite and incontrovertible lower limit: 0 K. Among the closest things to absolute zero in the universe is the temperature of supermassive black holes (10-18 K). At this temperature it will take them 10100 years and more to evaporate their mass. Yes, that's a one with one-hundred zeros. If the universe really does keep on expanding as believed by most scientist today, supermassive black holes will be the last remaining objects in the fading universe. Compared to their temperature, the lowest temperature ever achieved in a laboratory (10-12 K) is a true hellfire, despite it being many orders of magnitudes lower than the background temperature of the universe (2.73 K and slowly decreasing).
In terms of temperature, helium is an exceptional element. The fact that we almost always find it in the gaseous state is a result of its low boiling point (4.22 K). Even on Uranus (53 K), since the downgrading of Pluto the coldest planet in the solar system and by far the planet with the most inappropriate name, it would appear as a gas. Another temperature you definitely should remember is 92 K. Why? Because at this temperature the material Y-Ba-Cu-oxide becomes superconductive and there is no material known to man that is superconductive at higher temperatures. Note that you want a superconductor to do what it does best at temperatures as close to room temperature as possible because otherwise making use of this effect will require enormous amounts of energy for cooling.
The lowest officially recorded air temperature on Earth is 184 K ≈ -89 °C, so measured in 1983 in Stántsiya Vostók, Antarctica. Just recently scientists reported seeing an even lower temperature, but at the time of writing this is still unconfirmed. The next two values are very familiar to you: the melting point (273 K ≈ 0 °C) and the boiling point (373 K ≈ 100 °C) of water. But I would not advise you to become too familiar with burning wood (1170 K ≈ 900 °C) or the surface of our Sun (5780 K ≈ 5500 °C).
Temperatures in a lightning channel can go far beyond that, up to about 28,000 K. This was topped on August 6, 1945, when the atomic bomb "Little Boy" was dropped on Hiroshima. It is estimated that at a distance of 17 meters from the center of the blast the temperature rose to 300,000 K. Later and more powerful models of the atomic bomb even went past the temperature of the solar wind (800,000 K).
If you are disappointed about the relatively low surface temperature of the sun, keep in mind that this is the coldest part of the sun. In the corona surrounding it, temperatures can reach 10 million K, the center of the Sun is estimated to be at 16 million K and solar flares can be as hot as 100 million K. Surprisingly, mankind managed to top that. The plasma in the experimental Tokamak Fusion Test Reactor was recorded at mind-blowing 530 million K. Except for supernova explosions (10 billion K) and infant neutron stars (1 trillion K), there's not much beyond that.
One sunny day we arrive at work in the university administration to find a lot of aggressive emails in our in‒box. Just the day before, a news story about gender discrimination in academia was published in a popular local newspaper which included data from our university. The emails are a result of that. Female readers are outraged that men were accepted at the university at a higher rate, while male readers are angry that women were favored in each course the university offers. Somewhat puzzled, you take a look at the data to see what's going on and who's wrong.
The university only offers two courses: physics and sociology. In total, 1000 men and 1000 women applied. Here's the breakdown:
Seems like the male readers are right. In each course women were favored. But why the outrage by female readers? Maybe they focused more on the following piece of data. Let's count how many men and women were accepted overall.
Wait, what? How did that happen? Suddenly the situation seems reversed. What looked like a clear case of discrimination of male students turned into a case of discrimination of female students by simple addition. How can that be explained?
The paradoxical situation is caused by the different capacities of the two departments as well as the student's overall preferences. While the physics department, the top choice of male students, could accept 560 students, the smaller sociology department, the top choice of female students, could only take on 400 students. So a higher acceptance rate of male students is to be expected even if women are slightly favored in each course.
While this might seem to you like an overly artificial example to demonstrate an obscure statistical phenomenon, I'm sure the University of California (Berkeley) would beg to differ. It was sued in 1973 for bias against women on the basis of these admission rates:
A further analysis of the data however showed that women were favored in almost all departments ‒ Simpson's paradox at work. The paradox also appeared (and keeps on appearing) in clinical trials. A certain treatment might be favored in individual groups, but still prove to be inferior in the aggregate data.
In theory, hitting the middle C on a piano should produce a sound wave with a frequency of 523.25 Hz and nothing else. However, running the resulting audio through a spectrum analyzer, it becomes obvious that there's much more going on. This is true for all other instruments, from tubas to trumpets, basoons to flutes, contrabasses to violins. Play any note and you'll get a package of sound waves at different frequencies rather than just one.
First of all: why is that? Let's focus on stringed instruments. When you plug the string, it goes into its most basic vibration mode: it moves up and down as a whole at a certain frequency f. This is the so called first harmonic (or fundamental). But shortly after that, the nature of the vibration changes and the string enters a second mode: while one half of the string moves up, the other half moves down. This happens naturally and is just part of the string's dynamics. In this mode, called the second harmonic, the vibration accelerates to a frequency of 2 * f. The story continues in this fashion as other modes of vibration appear: the third harmonic at a frequency 3 * f, the fourth harmonic at 4 * f, and so on.
A note is determined by the frequency. As already stated, the middle C on the piano should produce a sound wave with a frequency of 523.25 Hz. And indeed it does produce said sound wave, but it is only the first harmonic. As the string continues to vibrate, all the other harmonics follow, producing overtones. In the picture below you can see which notes you'll get when playing a C (overtone series):
Quite the package! And note that the major chord is fully included within the first four overtones. So it's buy a note, get a chord free. And unless you digitally produce a note, there's no avoiding it. You might wonder why it is that we don't seem to perceive the additional notes. Well, we do and we don't. We don't perceive the overtones consciously because the amplitude, and thus volume, of each harmonic is smaller then the amplitude of the previous one (however, this is a rule of thumb and exceptions are possible, any instrument will emphasize some overtones in particular). But I can assure you that when listening to a digitally produced note, you'll feel that something's missing. It will sound bland and cold. So unconsciously, we do perceive and desire the overtones.
If you're not interested in mathematics, feel free to stop reading now (I hope you enjoyed the post so far). For all others: let's get down to some mathematical business. The frequency of a note, or rather of its first harmonic, can be computed via:
(1) f(n) = 440 * 2n/12
With n = 0 being the chamber pitch and each step of n one half-tone. For example, from the chamber pitch (note A) to the middle C there are n = 3 half-tone steps (A#, B, C). So the frequency of the middle C is:
f(3) = 440 * 23/12 = 523.25 Hz
As expected. Given a fundamental frequency f = F, corresponding to a half-step-value of n = N, the freqency of the k-th harmonic is just:
(2) f(k) = k * F = k * 440 * 2N/12
Equating (1) and (2), we get a relationship that enables us to identify the musical pitch of any overtone:
440 * 2n/12 = k * 440 * 2N/12
2n/12 = k * 2N/12
n/12 * ln(2) = ln(k) + N/12 * ln(2)
n/12 = ln(k)/ln(2) + N/12
(3) n – N = 12 * ln(k) / ln(2) ≈ 17.31 * ln(k)
The equation results in this table:
k
n – N (rounded)
1
0
2
12
3
19
4
24
5
28
And so on. How does this tell us where the overtones are? Read it like this:
The first harmonic (k = 1) is zero half-steps from the fundamental (n-N = 0). So far, so duh.
The second harmonic (k = 2) is twelve half-steps, or one octave, from the fundamental (n-N = 12).
The third harmonic (k = 3) is nineteen half-steps, or one octave and a quint, from the fundamental (n-N = 19).
The fourth harmonic (k = 4) is twenty-four half-steps, or two octaves, from the fundamental (n-N = 24).
The fifth harmonic (k = 5) is twenty-wight half-steps, or two octaves and a third, from the fundamental (n-N = 28).
So indeed the formula produces the correct overtone series for any note. And for any note the same is true: The second overtone is exactly one octave higher, the third harmonic one octave and a quint higher, and so on. The corresponding major chord is always contained within the first five harmonics.
The siren of an approaching police car will sound at a higher pitch, the light of an approaching star will be shifted towards blue and a passing supersonic jet will create a violent thunder. What do these phenomenon have in common? All of them are a result of the Doppler effect. To understand how it arises, just take a look at the animations below.
Stationary Source: The waves coming from the source propagate symmetrically.
Recently I posted a short introduction to recurrence relations – what they are and how they can be used for mathematical modeling. This post expands on the topic as car-following models are a nice example of recurrence relations applied to the real-world.
Suppose a car is traveling on the road at the speed u(t) at time t. Another car approaches this car from behind and starts following it. Obviously the driver of the car that is following cannot choose his speed freely. Rather, his speed v(t) at time t will be a result of whatever the driver in the leading car is doing.
The most basic car-following model assumes that the acceleration a(t) at time t of the follower is determined by the difference in speeds. If the leader is faster than the follower, the follower accelerates. If the leader is slower than the follower, the follower decelerates. The follower assumes a constant speed if there's no speed difference. In mathematical form, this statement looks like this:
a(t) = λ * (u(t) – v(t))
The factor λ (sensitivity) determines how strongly the follower accelerates in response to a speed difference. To be more specific: it is the acceleration that results from a speed difference of one unit.
——————————————
Before we go on: how is this a recurrence relation? In a recurrence relation we determine a quantity from its values at an earlier time. This seems to be missing here. But remember that the acceleration is given by:
a(t) = (v(t+h) – v(t)) / h
with h being a time span. Inserted into the above car-following equation, we can see that it indeed implies a recurrence relation.
——————————————
Our model is still very crude. Here's the biggest problem: The response of the driver is instantaneous. He picks up the speed difference at time t and turns this information into an acceleration also at time t. But more realistically, there will be a time lag. His response at time t will be a result of the speed difference at an earlier time t – Λ, with Λ being the reaction time.
a(t) = λ * (u(t – Λ) – v(t – Λ))
The reaction time is usually in the order of one second and consist of the time needed to process the information as well as the time it takes to move the muscles and press the pedal. There are several things we can do to make the model even more realistic. First of all, studies show that the speed difference is not the only factor. The distance d(t) between the leader and follower also plays an important role. The smaller it is, the stronger the follower will react. We can take this into account by putting the distance in the denominator:
a(t) = (λ / d(t)) * (u(t – Λ) – v(t – Λ))
You can also interpret this as making the sensitivity distance-dependent. There's still one adjustment we need to make. The above model allows any value of acceleration, but we know that we can only reach certain maximum values in a car. Let's symbolize the maximum acceleration by a(acc) and the maximum deceleration by a(dec). The latter will be a number smaller than zero since deceleration is by definition negative acceleration. We can write:
Recurrence relations are a powerful tool for mathematical modeling and numerically solving differential equations (no matter how complicated). And as luck would have it, they are relatively easy to understand and apply. So let's dive right into it using a purely mathematical example (for clarity) before looking at a real-world application.
This equation is a typical example of a recurrence relation:
x(t+1) = 5 * x(t) + 2 * x(t-1)
At the heart of the equation is a certain quantity x. It appears three times: x(t+1) stands for the value of this quantity at a time t+1 (next month), x(t) for the value at time t (current month) and x(t-1) the value at time t-1 (previous month). So what the relation allows us to do is to determine the value of said quantity for the next month, given that we know it for the current and previous month. Of course the choice of time span here is just arbitrary, it might as well be a decade or nanosecond. What's important is that we can use the last two values in the sequence to determine the next value.
Suppose we start with x(0) = 0 and x(1) = 1. With the recurrence relation we can continue the sequence step by step:
x(2) = 5 * x(1) + 2 * x(0) = 5 * 1 + 2 * 0 = 5
x(3) = 5 * x(2) + 2 * x(1) = 5 * 5 + 2 * 1 = 27
x(4) = 5 * x(3) + 2 * x(2) = 5 * 27 + 2 * 5 = 145
And so on. Once we're given the "seed", determining the sequence is not that hard. It's just a matter of plugging in the last two data points and doing the calculation. The downside to defining a sequence recursively is that if you want to know x(500), you have to go through hundreds of steps to get there. Luckily, this is not a problem for computers.
In the most general terms, a recurrence relation relates the value of quantity x at a time t + 1 to the values of this quantity x at earlier times. The time itself could also appear as a factor. So this here would also be a legitimate recurrence relation:
x(t+1) = 5 * t * x(t) – 2 * x(t-10)
Here we calculate the value of x at time t+1 (next month) by its value at a time t (current month) and t – 10 (ten months ago). Note that in this case you need eleven seed values to be able to continue the sequence. If we are only given x(0) = 0 and x(10) = 50, we can do the next step:
x(11) = 5 * 10 * x(10) – 2 * x(0) = 5 * 10 * 50 – 2 * 0 = 2500
But we run into problems after that:
x(12) = 5 * 11 * x(11) – 2 * x(1) = 5 * 11 * 2500 – 2 * x(1) = ?
We already calculated x(11), but there's nothing we can do to deduce x(1).
Now let's look at one interesting application of such recurrence relations, modeling the growth of animal populations. We'll start with a simple model that relates the number of animals x in the next month t+1 to the number of animals x in the current month t as such:
x(t+1) = x(t) + f * x(t)
The factor f is a constant that determines the rate of growth (to be more specific: its value is the decimal percentage change from one month to the next). So if our population grows with 25 % each month, we get:
x(t+1) = x(t) + 0.25 * x(t)
If we start with x(0) = 100 rabbits at month t = 0 we get:
x(1) = x(0) + 0.1 * x(0) = 100 + 0.25 * 100 = 125 rabbits
x(2) = x(1) + 0.1 * x(1) = 125 + 0.25 * 125 = 156 rabbits
x(3) = x(2) + 0.1 * x(2) = 156 + 0.25 * 156 = 195 rabbits
x(4) = x(3) + 0.1 * x(3) = 195 + 0.25 * 195 = 244 rabbits
x(5) = x(4) + 0.1 * x(4) = 244 + 0.25 * 244 = 305 rabbits
And so on. Maybe you already see the main problem with this exponential model: it just keeps on growing. This is fine as long as the population is small and the environment rich in ressources, but every environment has its limits. Let's fix this problem by including an additional term in the recurrence relation that will lead to this behavior:
- Exponential growth as long as the population is small compared to the capacity
– Slowing growth near the capacity
– No growth at capacity
– Population decline when over the capacity
How can we translate this into mathematics? It takes a lot of practice to be able to tweak a recurrence relation to get the behavior you want. You just learned your first chord and I'm asking you to play Mozart, that's not fair. But take a look at this bad boy:
x(t+1) = x(t) + a * x(t) * (1 – x(t) / C)
This is called the logistic model and the constant C represents said capacity. If x is much smaller than the capacity C, the ratio x / C will be close to zero and we are left with exponential growth:
x(t+1) ≈ x(t) + a * x(t) * (1 – 0)
x(t+1) ≈ x(t) + a * x(t)
So this admittedly complicated looking recurrence relation fullfils our first demand: exponential growth for small populations. What happens if the population x reaches the capacity C? Then all growth should stop. Let's see if this is the case. With x = C, the ratio x / C is obviously equal to one, and in this case we get:
x(t+1) = x(t) + a * x(t) * (1 – 1)
x(t+1) = x(t)
The number of animals remains constant, just as we wanted. Last but not least, what happens if (for some reason) the population gets past the capacity, meaning that x is greater than C? In this case the ratio x / C is greater than one (let's just say x / C = 1.2 for the sake of argument):
x(t+1) = x(t) + a * x(t) * (1 – 1.2)
x(t+1) = x(t) + a * x(t) * (- 0.2)
The second term is now negative and thus x(t+1) will be smaller than x(t) – a decline back to capacity. What an enormous amount of beautiful behavior in such a compact line of mathematics! This is where the power of recurrence relations comes to light. Anyways, let's go back to our rabbit population. We'll let them grow with 25 % (a = 0.25), but this time on an island that can only sustain 300 rabbits at most (C = 300). Thus the model looks like this:
x(t+1) = x(t) + 0.25 * x(t) * (1 – x(t) / 300)
If we start with x(0) = 100 rabbits at month t = 0 we get 153 + 0.25 * 153 * (1 – 153 / 300) = 172 rabbits
x(5) = 172 + 0.25 * 172 * (1 – 172 / 300) = 190 rabbits
Note that now the growth is almost linear rather than exponential and will slow down further the closer we get to the capacity (continue the sequence if you like, it will gently approach 300, but never go past it).
We can even go further and include random events in a recurrence relation. Let's stick to the rabbits and their logistic growth and say that there's a p = 5 % chance that in a certain month a flood occurs. If this happens, the population will halve. If no flood occurs, it will grow logistically as usual. This is what our new model looks like in mathematical terms:
x(t+1) = x(t) + 0.25 * x(t) * (1 – x(t) / 300) if no flood occurs
x(t+1) = 0.5 * x(t) if a flood occurs
To determine if there's a flood, we let a random number generator spit out a number between 1 and 100 at each step. If it displays the number 5 or smaller, we use the "flood" equation (in accordance with the 5 % chance for a flood). Again we turn to our initial population of 100 rabbits with the growth rate and capacity unchanged 0.5 * 153 = 77 rabbits
x(5) = 77 + 0.25 * 77 * (1 – 77 / 300) = 91 rabbits
As you can see, in this run the random number generator gave a number 5 or smaller during the fourth step. Accordingly, the number of rabbits halved. You can do a lot of shenanigans (and some useful stuff as well) with recurrence relations and random numbers, the sky's the limit. I hope this quick overview was helpful.
A note for the advanced: here's how you turn a differential equation into a recurrence relation. Let's take this differential equation:
dx/dt = a * x * exp(- b*x)
First multiply by dt:
dx = a * x * exp(- b * x) * dt
We set dx (the change in x) equal to x(t+h) – x(t) and dt (change in time) equal to a small constant h. Of course for x we now use x(t):
x(t+h) – x(t) = a * x(t) * exp(- b * x(t)) * h
Solve for x(t+h):
x(t+h) = x(t) + a * x(t) * exp(- b * x(t)) * h
And done! The smaller your h, the more accurate your numerical results. How low you can go depends on your computer's computing power.
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Department of Mathematics
Overview In a world of increasing technological
complexity, knowledge of mathematics is the gateway to the pursuit
of many fields. Mathematics has long been the language of choice
for expressing complex relationships and describing complicated
patterns and processes. It is now true that many fields, in addition
to mathematics and the sciences, rely on this in a fundamental way.
What was formerly "abstract" mathematics to many has become the
concrete stuff of everyday life. "The unreasonable effectiveness
of mathematics" manifests itself today in such familiar things as
CAT and MRI scans, compact discs, satellite communications, and
computer animation. These were all rendered possible by new discoveries
made by mathematicians within the last fifty years. Even the efficient
operation of our financial markets is based, in part, on relatively
recent theorems of mathematical analysis and probability theory.
Mathematics research today is a vibrant and dynamic enterprise.
Thousands of mathematicians worldwide are at work on an unimaginably
broad range of questions. Exciting recent advances include the proof
of Fermat's Last Theorem, the classification of the finite simple
groups, the proof of the Bieberbach conjecture, and the computer-assisted
proof of the four-color theorem. The discipline and creativity required
by the study of mathematics can be a formidable preparation for
later life. Past students of mathematics have had successful careers
in almost every sphere, including all the professions. The scope
of mathematics courses offered at the University of Virginia allows
majors to tailor their own programs. Students electing to major
in mathematics should consult carefully with a faculty advisor to
ensure the selection of a program of courses that provides a solid
grounding in the fundamentals of higher mathematics and is appropriate
to future goals.
Faculty The faculty of the Department of Mathematics
is committed to excellence in teaching and research. Its members
carry out high-level research on diverse problems in algebra, analysis,
topology, probability, and statistics, mathematical physics, and
the history of the discipline. Their research has been widely published
in prestigious research journals and is recognized internationally.
Members of the department have won Sloan fellowships, Humboldt fellowships,
and other scholarly honors, as well as numerous research grants.
Many are currently supported by grants from the National Science
Foundation and other federal agencies. Most have held visiting professorships
abroad. In addition, the department offerings and ambiance are enhanced
each year by the presence of several internationally recognized
visiting faculty.
Students There are currently about 75 students
majoring in mathematics. Class sizes vary from a few large introductory
classes to an average class size of twenty students for upper-level
courses. This small class size affords students the opportunity
to get individual attention.
Students who graduate with degrees in mathematics successfully
pursue a variety of different careers. Many go directly into jobs
in industry, insurance (as actuaries), government, finance, and
other fields. Employers in the past have included Morgan Stanley,
General Motors, MITRE Corp., the Census Bureau, the National Security
Agency, and various consulting firms. Many find themselves well-equipped
to go on to professional schools in law, medicine, and business.
Some go directly into teaching. Others have gone on to graduate
programs in mathematics, applied mathematics, statistics, engineering,
systems engineering, economics, and computer science. Students who
have combined the mathematics major with courses in computer programming,
economics, and business have done exceptionally well in the job
market.
Requirements for Major Normally, the calculus
sequence MATH 131, 132, and 231 or its equivalent must be completed
before a student can declare a major in mathematics. At least a
2.2 average in the calculus sequence and a minimum grade of C in
MATH 231 or its equivalent are required. However, the department
may grant special permission to declare a major to a student who
has only completed MATH 131 and 132, and at least one mathematics
course (other than MATH 231 or its equivalent) which could be counted
toward the major in mathematics, provided the student completes
MATH 231 or its equivalent in the semester following the declaration
of a mathematics major.
To graduate with a major in mathematics the student must show computer
proficiency by completing CS 101 or 120, or an approved equivalent
course. This should be done as early as possible.
To help guide the student through the major, the mathematics department
offers six concentrations. Completion of one of these concentrations
is required. Each concentration contains a set of nine or ten required
courses (approximately 28 credit hours). To graduate, a student
must obtain minimum grades of C in seven of these courses and C-
in the other two.
Certain substitutions are allowed in all options, for example,
MATH 531 for MATH 331 and MATH 551 for MATH 354.
A. The Basic Concentration
This traditional program for the mathematics major provides an overview
of key areas:
MATH
325
Ordinary
Differential Eq.
4
MATH
351
Elementary
Linear Algebra
3
MATH
354
Survey
of Algebra
3
Two
from the following three:
MATH
310
Introduction
to
Mathematical Probability
3
MATH
331
Basic
Real Analysis
3
MATH
334
Complex
Variables with Applications
3
Four
electives at the 300 level or higher
12
Students fulfilling the requirements for this option have a wide
range of career opportunities, from law to business to any field
that requires deductive, logical reasoning skills.
B. The Graduate Preparatory Concentration
This concentration is for the student who plans to attend graduate
school in mathematics or an allied field. The program emphasizes
the fundamental ideas of mathematics with substantial work in proving
and understanding the basic theorems. It consists of:
MATH
325
Ordinary
Differential Eq.
4
MATH
334
Complex
Variables with Applications
3
MATH
351
Elementary
Linear Algebra
3
MATH
531
Intro.
to Real Analysis I
3
MATH
551
Intro.
to Abstract Algebra I
3
MATH
552
Intro.
to Abstract Algebra II
3
Three electives at the 300 level or higher.(You may wish to take
MATH 354 in preparation for MATH 551 andáMATH 331 in preparation
for MATH 531.)
This constitutes the minimum expected of an incoming graduate student
in most programs nationwide. The department strongly recommends
MATH 532 (Real Analysis in Several Variables), as well as courses
in differential geometry and topology (MATH 572 and 577). Many of
our graduate school bound students take additional courses, including
700-level graduate courses.
C. The Probability and Statistics Concentration This concentration
is designed to give the student a good theoretical underpinning
in probability and statistics, as well as the opportunity to go
deeper in these fields. The program can lead to a Master of Science
in Statistics with one additional year of course work, if additional
courses in statistics are taken in the fourth year. (Those interested
in the M.S. in Statistics should contact the graduate advisor in
the Department of Statistics prior to the beginning of their fourth
year.) The requirements for the concentration are the following:
MATH
325
Ordinary
Differential Eq.
4
MATH
310
Intro.
to Mathematical
Prob511
Stochastic
Processes
3
STAT
512
Applied
Linear Models
3
One
additional course chosen from:
MATH
430
Elementary
Numerical
Analysis
3
MATH
531
Intro.
to Real Analysis I
3
STAT
313
Design
and Analysis of Sample Surveys
3
STAT
513
Applied
Multivariate Statistics
3
STAT
516
Experimental
Design
3
STAT
517
Applied
Time Series
3
STAT
519
Intro.
to Mathematical
Statistics
3
D. The Financial Mathematics Concentration This program
provides the student with a broad background of basic mathematics
which is essential for an understanding of the mathematical models
used in the financial markets. The mathematics of modern finance
includes, but is not limited to, probability, statistics, regression,
time series, partial differential equations, stochastic processes,
stochastic calculus, numerical methods, and analysis. Probability
and statistics and some acquaintance with numerical methods are
essential as is some knowledge of economics/accounting and some
computing experience. Additional background in statistics, optimization,
and stochastic processes is also desirable. The program consists
of:
E. Actuarial Concentration This concentration offers some
of the basic mathematics and statistics necessary for a successful
career in actuarial science, and it provides some of the academic
background needed to pass the first few actuarial exams.
Actuaries use mathematics, statistics, and financial theory to
analyze future events, especially those related to insurance and
pension programs. They may work for insurance companies, consulting
firms, government, employee benefits departments of large organizations,
banks, investment firms, or more generally, businesses that need
to assess the financial consequences of risk.
To become an actuary, the student must pass a series of examinations
administered by the professional actuarial societies: the Society
of Actuaries (SOA) and the Casualty Actuarial Society (CAS). The
first few exams are jointly administered. Exams which correspond
to various courses are indicated below.
It is highly advantageous for students interested in this concentration
to take both MATH 310 and 312 in their second year. Actuarial Mathematics
(MATH 517/STAT 540) and Actuarial Risk Theory (STAT 541) form the
core of the actuarial program. Both of these courses are offered
every year if there is sufficient student interest, and otherwise
in alternate years. With sufficient early course preparation, a
summer internship after the third year has been an integral part
of the program for those students who wished to intern.
F. Five-year Teacher Education Program This option leads
to both Bachelor of Arts and Master of Teaching degrees after five
years. The program is for both elementary and secondary teachers
and is administered by the Curry School of Education. Required courses
include:
MATH
325
Ordinary
Differential Equations
4
MATH
310
Intro.
to Mathematical Prob501
History
of Calculus or
MATH
503
History
of Mathematics
3
MATH
570
Introduction
to Geometry
3
One
elective at the 300 level or higher
3
The Curry School has additional requirements for this program.
Distinguished Majors Program in Mathematicsá The
department offers a Distinguished Majors Program (DMP) to qualified
majors in mathematics. Admission to the program is granted by the
departmental committee for the DMP, usually at the end of the student's
fourth semester. Criteria for acceptance into the program are based
on the GPA in mathematics, letters of recommendation from mathematics
instructors, and the cumulative GPA in the College (which should
be near 3.4 or higher).
The DMP is the same as the graduate school preparatory concentration,
except that in the fourth year the students also take the seminar
course MATH 583 in which they give an hour lecture and prepare a
written exposition of their work in the seminar under faculty guidance.
Note that MATH 531 and 551 are prerequisites for the seminar. As
with the concentrations, the DMP must consist of at least nine courses.
Three levels of distinction are possible: distinction, high distinction,
or highest distinction. The departmental recommendation for the
level of distinction to be awarded is based on the quality of the
student's seminar presentations, the overall work in the DMP, and
the entire major program, as well as the student's College GPA.
Requirements for Minor in Mathematicsá Students
who wish to declare a minor in mathematics must complete the calculus
sequence through MATH 231 or its equivalent with at least a 2.0
average.
To graduate with a minor in mathematics a student must complete
five courses approved by the department of mathematics with minimum
grades of C in three of the courses and minimum grades of C- in
the other two. An approved course must carry at least three credits.
Currently, the approved courses are those from the College department
of mathematics with the MATH mnemonic numbered 300 or higher. Courses
with the STAT mnemonic or from other departments or institutions
can be taken if approved by the undergraduate committee.
Courses that are being counted for a major or another minor cannot
also be counted for the minor in mathematics.
Echols Mathematics Club is an undergraduate club
for mathematics students that sponsors lectures, mathematics films,
problem solving sessions for the Putnam Mathematical Competition
and other similar activities.
Course Descriptions
The entering College student has a variety of courses in mathematics
from which to choose. Among those that may be counted toward the
College area requirement in natural science and mathematics, are
several options in calculus, elementary (non-calculus based) courses
in probability and in statistics, and courses dealing with computer
techniques in mathematics.
MATH 103 (precalculus) is available for students who need to improve
basic skills that are required in other courses such as calculus,
chemistry, psychology, economics, and statistics. However, it may
not be counted toward the area requirement in natural science and
mathematics. Students planning to major in the social sciences,
arts, or humanities who wish to take a mathematics course but omit
the study of calculus may choose from MATH 108 (Modes of Mathematical
Thinking) or MATH 111 (Elementary Probability Theory). Even though
it is not a prerequisite for STAT 112, MATH 111 is frequently taken
prior to STAT 112. MATH 115 and 116 are introductory courses that
investigate familiar areas of elementary mathematics at a profound
level and are intended for first- and second-year non-majors, especially
those preparing to teach in elementary and middle schools.
In MATH 114, the students study the mathematics needed to understand
and answer a variety of questions that arise in everyday financial
dealings. The emphasis in this course will be on applications, including
simple and compound interest, valuation of bonds, rates of return
on investments, and more. Although the topics in this course are
drawn primarily from business and economics, students of all majors
are welcome and should find the applications interesting and relevant.
The study of calculus is the foundation of college mathematics
for students planning to major in mathematics or the physical sciences
or anticipating a career or graduate study in any of the natural
sciences, engineering, or applied social sciences (such as economics).
There are essentially two programs of study available in calculus:
MATH 121, 122 is a terminal one-year sequence intended for business,
biology, and social science majors; MATH 131, 132, 231 is the traditional
calculus sequence intended for students of mathematics and the natural
sciences, as well as for students intending to pursue graduate work
in the applied social sciences;
The MATH 121, 122 sequence is unacceptable as a prerequisite for
mathematics courses numbered 231 and above. Students anticipating
the need for higher mathematics courses such as MATH 325 (Differential
Equations) or MATH 310, 312 (Probability and Statistics) should
instead elect the MATH 131, 132, 231 sequence. Credit is not allowed
for both MATH 121 and 131 (or its equivalent).
Students who have previously passed a calculus course in high school
may elect MATH 122, 131, 132, or 231 as their first course, depending
on placement, preparation, and interest. A strong high school calculus
course is generally adequate preparation for MATH 132 as a first
calculus course, even if advanced placement credit has not been
awarded for MATH 131. Students planning to take any advanced course
in mathematics should not take MATH 122, because credit for that
course must be forfeited if the student takes MATH 132 (or its equivalent).
MATH 133 and 134 is a two semester calculus workshop sequence taken
in conjunction with specific sections of MATH 131 and 132. Participants
in the calculus workshop meet for six hours per week to work in
small groups on challenging problem sets related to material covered
in MATH 131 and 132. They typically enjoy getting to work closely
with fellow calculus students, and find that their performance in
MATH 131 and 132 is significantly improved. Permission is required
to sign up for the calculus workshop. For more information, contact
Professor Jeffrey Holt, Calculus Workshop Coordinator; 924-4927;
jjh2b@virginia.edu.
Exceptionally well prepared students (who place out of both MATH
131 and 132) may choose either MATH 231 or 325 (Differential Equations)
as their first course.
Advanced placement credit in the calculus sequence is granted on
the basis of the College Entrance Examination Board Advanced Placement
Test (either AB or BC). A score of 4 or 5 on the AB test or on the
AB subscore of the BC test gives the student credit for MATH 131.
A score of 4 or 5 on the BC test gives the student credit for both
MATH 131 and 132. Students who wish to enter the calculus sequence
but who have not received advanced placement credit should consult
the Student Handbookáfor placement guidelines based on grades and
achievement test scores. The Department of Mathematics offers short
advisory placement tests during fall orientation.
Pre-commerce students are required to take a statistics course,
usually STAT 112, and one other mathematics course, usually MATH
111, 121, 122, or MATH 131.
Warning áThere are numerous instances of equivalent
courses offered by the Department of Mathematics as well as by the
Department of Applied Mathematics in the School of Engineering and
Applied Science. A student may not offer for degree credit two equivalent
courses (e.g., MATH 131 and APMA 101, or MATH 131 and MATH 121).
MATH 108 - (3) (IR)
Modes of Mathematical Thinking
Studies logic, number systems, functions, analytic geometry, equations,
matrices, enumeration, computer algebra systems. Intended for liberal
arts students and emphasizes the connection between analytic-algebraic
and geometric reasoning in the understanding of mathematics. Facilitated
by the use of a modern computer algebra system, such as Maple.
MATH 114 - (3) (Y)
Financial Mathematics
The study of the mathematics needed to understand and answer a variety
of questions that arise in everyday financial dealings. The emphasis
is on applications, including simple and compound interest, valuation
of bonds, amortization, sinking funds, and rates of return on investments.
A solid understanding of algebra is assumed.
MATH 115 - (3) (IR)
The Shape of Space
Provides an activity and project-based exploration of informal geometry
in two and three dimensions. Emphasizes visualization skill, fundamental
geometric concepts, and the analysis of shapes and patterns. Topics
include concepts of measurement, geometric analysis, transformations,
similarity, tessellations, flat and curved spaces, and topology.
MATH 116 - (3) (IR)
Algebra, Number Systems, and Number Theory
Studies basic concepts, operations, and structures occurring in
number systems, number theory, and algebra. Inquiry-based student
investigations explore historical developments and conceptual transitions
in the development of number and algebraic systems.
MATH 121 - (3) (S)
Applied Calculus I
Topics include limits and continuity; differentiation and integration
of algebraic and elementary transcendental functions; and applications
to maximum-minimum problems, curve sketching and exponential growth.
Credit is not given for both MATH 121 and 131.
MATH 122 - (3) (S)
Applied Calculus II Prerequisite:áMATH 121 or equivalent.
A second calculus course for business, biology, and social science
students. Analyzes functions of several variables, their graphs,
partial derivatives and optimization; multiple integrals. Reviews
basic single variable calculus and introduces differential equations
and infinite series. Credit is not given for both MATH 122 and 132.
MATH 131 - (4) (S)
Calculus I Prerequisite: Background in algebra, trigonometry, exponentials,
logarithms, and analytic geometry.
Introduces calculus with emphasis on techniques and applications.
Recommended for natural science majors and students planning additional
work in mathematics. The differential and integral calculus for
functions of a single variable is developed through the fundamental
theorem of calculus. Credit is not given for both MATH 121 and 131.
MATH 132 - (4) (S)
Calculus II Prerequisite:áMATH 131 or equivalent, or instructor permission.
Continuation of 131. Applications of the integral, techniques of
integration, infinite series, vectors. Credit is not given for both
MATH 122 and 132.
MATH 231 - (4) (S)
Calculus III Prerequisite:áMATH 132 or its equivalent.
Studies functions of several variables including lines and planes
in space, differentiation of functions of several variables, maxima
and minima, multiple integration, line integrals, and volume.
MATH 300 - (3) (IR)
Foundations of Analysis Prerequisite:áMATH 132 or equivalent.
Topics from logic and the construction of mathematical proofs, basic
set theory, number systems, continuity of functions, and foundations
of analysis. Intermediate introduction of the standards of mathematical
rigor and abstraction that are encountered in advanced mathematics,
based on the material of the calculus and other basic mathematics.
MATH 325P - (4) (S)
Ordinary Differential Equations Prerequisite:áMATH 132 or its equivalent.
Usually offered in the spring, this course covers the same material
as MATH 325 with some additional topics, including an introduction
to Sturm-Liouville theory, Fourier series and boundary value problems,
and their connection with partial differential equations. Physics
majors should enroll in MATH 325P, although no knowledge of physics
is assumed.
MATH 331 - (3) (S)
Basic Real Analysis Prerequisite:áMATH 132.
Concentrates on proving the basic theorems of calculus, with due
attention to the beginner with little or no experience in the techniques
of proof. Includes limits, continuity, differentiability, the Bolzano-Weierstrass
theorem, Taylor's theorem, integrability of continuous functions,
and uniform convergence.
MATH 354 - (3) (Y)
Survey of Algebra Prerequisite:áMATH 132 or equivalent.
Surveys major topics of modern algebra: groups, rings, and fields.
Presents applications to areas such as geometry and number theory;
explores rational, real, and complex number systems, and the algebra
of polynomials.
MATH 475 - (3) (IR)
Introduction to Knot Theory Prerequisite:áMATH 331, 354, or instructor permission. Examines
the knotting and linking of curves in space. Studies equivalence
of knots via knot diagrams and Reidemeister moves in order to define
certain invariants for distinguishing among knots. Also considers
knots as boundaries of surfaces and via algebraic structures arising
from knots.
MATH 493 - (3) (IR)
Independent Study
Reading and study programs in areas of interest to individual students.
For third- and fourth- years interested in topics not covered in
regular courses. Students must obtain a faculty advisor to approve
and direct the program.
MATH 495 - (3) (IR)
Undergraduate Research Seminar Prerequisite: Instructor permission.
Emphasizes direct contact with advanced mathematical ideas, communication
of these ideas, the discovery of new results and connections among
them, and the experience of mathematics as a collaborative venture
among researchers at all levels. Students work collaboratively and
individually on research projects, and present their results to
the class.
MATH 501 - (3) (E)
The History of the Calculus Prerequisite:áMATH 231 and 351 or instructor permission.
Studies the evolution of the various mathematical ideas leading
up to the development of calculus in the 17th century, and how those
ideas were perfected and extended by succeeding generations of mathematicians.
Emphasizes primary source materials when possible.
MATH 503 - (3) (O)
The History of Mathematics Prerequisite:áMATH 231 and 351 or instructor permission.
Studies the development of mathematics from classical antiquity
to the end of the 19th century, focusing on critical periods in
the evolution of geometry, number theory, algebra, probability,
and set theory. Emphasizes primary source materials when possible.
MATH 510 - (3) (Y)
Mathematical Probability Prerequisite: Graduate standing and MATH 132, or equivalent.
Those who have received credit for MATH 310 may not take 510 for
credit. Studies the development and analysis of probability models
through the basic concepts of sample spaces, random variables, probability
distributions, expectations, and conditional probability. Also includes
distributions of transformed variables, moment generating functions,
and the central limit theorem.
MATH 512 - (3) (Y)
Mathematical Statistics Prerequisite:áMATH 510 and graduate standing.
Topics include methods of estimation, general concepts of hypothesis
testing, linear models and estimation by least squares, categorical
data, and nonparametric statistics. Those who have received credit
for MATH 312 may not take 512 for credit.
MATH 514 - (3) (Y)
Mathematics of Derivative Securities Prerequisite:áMATH 231 or 122 and a knowledge of probability and
statistics. MATH 310 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards
and swaps, hedging, option-pricing theory, and sensitivity analysis.
MATH 517 - (3) (IR)
Actuarial Mathematics Prerequisite:áMATH 312 or 512, instructor permission.
Covers the main topics required by students preparing for the examinations
in actuarial statistics, set by the American Society of Actuaries.
Topics include life tables, life insurance and annuities, survival
distributions, net premiums and premium reserves, multiple life
functions and decrement models, valuation of pension plans, insurance
models, benefits, and dividends.
MATH 554 - (3) (Y)
Survey of Algebra Prerequisite:áMATH 132 or equivalent and graduate standing.
Surveys groups, rings, and fields, and presents applications to
other areas of mathematics, such as geometry and number theory.
Explores the rational, real, and complex number systems, and the
algebra of polynomials.
MATH 555 - (3) (IR)
Algebraic Automata Theory Prerequisite:áMATH 351.
Introduces the theory of sequential machines, including an introduction
to the theory of finite permutation groups and transformation semigroups.
Includes examples from biological and electronic systems as well
as computer science, the Krohn-Rhodes decomposition of a state machine,
and Mealy machines.
MATH 572 - (3) (IR)
Introduction to Differential Geometry Prerequisite:áMATH 231.
Topics selected by the instructor from the theory of curves and
surfaces in Euclidean space and the theory of manifolds.
MATH 596 - (3) (S)
Supervised Study in Mathematics Prerequisite: Instructor permission and graduate standing.
In exceptional circumstances, a student may undertake a rigorous
program of supervised study designed to expose the student to a
particular area of mathematics. Regular homework assignments and
scheduled examinations are required.
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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Product Description
Clear explanations, biblical applications, and plenty of helpful examples will help guide students through the challenge of succeeding in Algebra 2. With an emphasis on thinking and reasoning skills, discussions center upon quadratic equations, polynomials, complex numbers, and trigonometry. Operations, linear equations, linear relations, polynomial functions, radicals, complex numbers, inverse functions, probability & statistics, and other elements are also explored. Special "Algebra and Scripture", "Algebra around the World" and "Algebra in History" segments help bring a wider perspective to the subject. Chapters provide objectives, clear explanations, and plenty of exercises and review (including word problems). 653 pages, softcover. Reference tables, a glossary, and selected answers are provided in the back. 2nd Edition.
This resource is also known as Bob Jones Algebra 2 Student Text, Grade 11, 2nd Edition with Updated Copyright.
When I bought this book I thought that it would be a good alternative to taking Algebra classes at a public school. But when I got it I was disapointed and confused. The way the material was presented was almost too challenging, especially for the slow learners. My advice is to only buy this material for advanced students or those who understand math very easily
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Prealgebra (Cloth) - 6th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. ''Prealgebra,'' Sixth Edition was written to help students effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success. Whole Numbers and Introduction to Algebra; Intege...show morers and Introduction to Solving Equations; Solving Equations and Problem Solving; Fractions and Mixed Numbers; Decimals; Ratio, Proportion, and Triangle Applications; Percent; Graphing and Introduction to Statistics; Geometry and Measurement; Exponents and Polynomials For all readers interested in prealgebra. ...show less
Ships next business day! May NOT include supplemental materials such as CDs and access codes. May include some highlighting or writing.
$27.272009 Hardcover Fair CONTAINS SLIGHT WATER DAMAGE / STAIN, STILL VERY READABLE, SAVE! This item may not include any CDs, Infotracs, Access cards or other supplementary material.
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newrecycleabook centerville, OH
032164008X used book - book appears to be recovered - has some used book stickers - free tracking number with every order. book may have some writing or highlighting, or used book stickers on front o...show morer
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McDougalLittellAlgebra1 *Destination Math does not align to all standards. Those standards are not shown on this document. This document is a correlation of Destination Math, to McDougalLittellAlgebra1. 2
Later chapters in McDougalLittell Math Course 1 include topics such as integers, functions, and probability. ... earlier grades with Algebra topics such as patterns variables, andformulas. AlgebraAlgebra Pre-Course Review • make a model, 761
McDougalLittell Pre-Algebrawill give you a strong foundation in algebra while also preparing you for future study of geometry, probability, and data analysis. The clearly written lessons make even difficult math concepts and methods understandable by providing numerous
Algebra1 M117 Course Syllabus • Algebra1 M117 is designed to expand your knowledge of Algebra in ways you can relate the concepts ... Textbook: Algebra1, McDougalLittell:Illinois Edition • Students will be marked tardy if they are not in their seat when the bell rings.
McDougalLittellAlgebra1 is the textbook that is used as a reference for this course. The textbook can be found online at: ndex.cfm?state=NJ . ALGEBRA1 . Grade 8 Algebra1 is a course designed for 8th grade
Two Algebra1 Word Problems Harris Shultz California ... (McDougalLittellAlgebra1 – Page 577): The glass has a height-to-width ratio of 3 : 2. The frame adds 6 ... Discussion: This is an illustrated example in the textbook, and the given solution begins by stating that the window has a ...
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Rates of Change and Limits Calculating Limits Using the Limit Laws Precise Definition of a Limit One-Sided Limits and Limits at Infinity Infinite Limits and Vertical Asymptotes Continuity Tangents and Derivatives
3. Differentiation
The Derivative as a Function Differentiation Rules The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule and Parametric Equations Implicit Differentiation Related Rates Linearization and Differentials
Estimating with Finite Sums Sigma Notation and Limits of Finite Sums The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Substitution Rule Substitution and Area Between Curves
6. Applications of Definite Integrals
Volumes by Slicing and Rotation About an Axis Volumes by Cylindrical Shells Lengths of Plane Curves Moments and Centers of Mass Areas of Surfaces of Revolution and The Theorems of Pappus Work Fluid Pressures and Forces
Sequences Infinite Series The Integral Test Comparison Tests The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series; Error Estimates Applications of Power Series Fourier Series
Double Integrals Areas, Moments and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals
16. Integration in Vector Fields
Line Integrals Vector Fields, Work, Circulation, and Flux Path Independence, Potential Functions, and Conservative Fields Green's Theorem in the Plane Surface Area and Surface Integrals Parametrized Surfaces Stokes' Theorem The Divergence Theorem and a Unified Theory Appendices Mathematical Induction Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers The Distributive Law for Vector Cross Products Determinants and Cramer's Rule The Mixed Derivative Theorem and the Increment Theorem The Area of a Parallelogram's Projection on a PlaneDream Books Company, LLC Englewood, CO
2007 HardcoverRF Ventures Booksellers Pleasant View, TN
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Calculus, An Introduction
Introduction
Integral calculus,
which measures the area of various shapes that don't have convenient,
straight-line boundaries,
began long before the birth of Christ.
Differential calculus, which deals with the instantaneous rate of change of a varying quantity,
did not arise until the 17th century.
Still, we present differential calculus first,
because it is easier to understand.
Then we develop integral calculus,
and finally we show how they are related to each other.
Most text books take this approach - but not all.
Calculus rests on a foundation of limits and sequences.
Some people never worry about this foundation,
and you can definitely get to the moon without it,
but if you want to be rigorous,
you should review the section on limits first.
Both Newton
(biography)
and Leibniz
(biography)
developed calculus independently,
using slightly different notation,
and argued over who deserved the credit for years after the fact.
| 677.169 | 1 |
Elementary Statistics - Text Only - 7th edition
Summary: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for general beginning statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. This edition features increased emphasis on Excel, MINITAB, and the TI-83 Plus graphing calculator, computing technologies commonly used in such coureses.
Great condition for a used book! Minimal wear. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!
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HPB-Cedar-Hill Cedar HillRockwall Rockwall 0073534978 Like New copy, without any marks or highlights. Has shelf wear on covers. Formula Card andCD included This is Student US Edition. A+ Customer Service!
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Nivea Books Lynnwood, WA
Hardcover Fine 0073534978 Like New copy, without any marks or highlights. CD AND FORMULAS CHART INCLUDED. Might have minor shelf wear on covers. This is Student US Edition. Same day shipping with f...show moreree tracking number. Expedited shipping available. A+ Customer Service15 +$3.99 s/h
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Big Planet Books Burbank, CA
2008-10-01 Hardcover Good Expedited shipping is available for this item!
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Haciendas De Borinquen Ii, PR Algebra they need to know what the problem or question is asking. Finally, identify what information is given or available in order to solve. Those who have tried this formula have found Math to be fun
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Linear Algebra with Applications
This book is for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite. This thorough and ...Show synopsisThis book is for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite. This thorough and accessible text, from one of the leading figures in the use of technology in linear algebra, gives students a challenging and broad understanding of the subject. The author infuses key concepts with their modern practical applications to offer students examples of how mathematics is used in the real world. Each chapter contains integrated worked examples and chapter tests. The book stresses the important roles geometry and visualization play in understanding linear algebra
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Understanding Elementary Algebra With Geometry A Course for College Students
9780534999728
0534999727
Summary: Dr. Arthur Goodman (Ph.D., Yeshiva University) currently teaches in the mathematics department at Queens College of the City University of New York.
Hirsch, Lewis is the author of Understanding Elementary Algebra With Geometry A Course for College Students, published 2005 under ISBN 9780534999728 and 0534999727. Three hundred five Understanding Elementary Algebra With Geometry A Course for College Students t...extbooks are available for sale on ValoreBooks.com, eighty one used from the cheapest price of $6.53, or buy new starting at $240 NO CD! ! This item may not inclu... [more]CONTAINS SLIGHT WATER DAMAGE / STAIN, STILL VERY READABLE, SAVE! NO CD! !GOOD - Clean and solid. Has some highlighting or writing. 100% Money Back Guarantee on all Items. We ship DAILY with free delivery confirmation. Choose expedited for FAST deli [more]
GOOD - Clean and solid. Has some highlighting or writing. 100% Money Back Guarantee on all Items. We ship DAILY with free delivery confirmation. Choose expedited for FAST delivery. We believe in providing accurate grading on used books and excellent customer service. Five Star Seller with thousands of satisfied customers... buy with confidence
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What
is Mathematics?
Mathematics is the study of the measurement, properties, and
relationships of quantities and sets, using numbers and symbols. It is a body of
related courses concerned with knowledge of measurement, properties, and
relations quantities, which can include theoretical or applied studies of
arithmetic, algebra, geometry, trigonometry, statistics, and calculus.
Mathematicians have an opportunity to make a lasting contribution to society
by helping to solve problems in such diverse fields as medicine, management,
economics, government, physics, and psychology.
Helpful Links
Your donations are greatly appreciated and help us to increase the support for students, faculty, academic programs, research, regional engagement, and global outreach to achieve excellence in all aspects of the UNCW experience.
| 677.169 | 1 |
investigate integral calculus. In this calculus lesson plan, students investigate the relationship between the area above and below the curve using integral calculus. Students look for a general rule for area above: area below.
Students explore integral calculus. For this calculus lesson, students explore the relationship between the area under the graph of a functions and the integral of the function. Students also discover the rule for the integral of f(x)=ax^n.
Students explore the Fundamental Theorem of Calculus. In the Calculus instructional activity, students investigate indefinite and definite integrals and the relationship between the two, which leads to the discovery of the Fundamental Theorem of Calculus.
Young scholars integrate integrals. In this integrating integrals lesson, students integrate definite and indefinite integrals. Young scholars use their Ti-89 to find the integrals of cubic, inverse, and trigonometric function graphs on a given intervalTwelfth graders solve first order differential equations using the separation of variables technique. In this calculus lesson, 12th graders explain the connection between math and engineering. They brainstorm what engineers do in real life.
Young scholars explore the concept of definite integrals. In this definite integrals lesson, students find the area between two curves. Young scholars use the Ti-Nspire to find the definite integrals of curves such as sine and cosine.
Students explore the area under the curve using the TI InterActive! document. In this calculus instructional activity, students input various functions and create tables. They answer questions based on their results.
Twelfth graders investigate the limitations of the Fundamental Theorem of Calculus. In this calculus lesson, 12th graders explore when one can and cannot use the Fundamental Theorem of Calculus and explore the definition of an improper integral.
Students use a graphing calculator to do calculations and edit their calculations using script. In this graphing calculator lesson, students compute the length of intervals given in parametric form. Students verify the results for specific given functions and define the critical points where the first derivative is found.
Students explore the concept of approximating integrals. In this approximating integrals lesson, students approximate integrals by finding the area of rectangles. Students use a graphing calculator to find the left, right, and midpoint riemann sums.
Learners investigate integral calculus. In this calculus instructional activity, students explore an application of integrations through a leaking hot tub problem. The activity emphasizes using the integral of a rate of change to give the accumulated change.
Students practice calculating and analyzing Riemann sums and illustrate when Riemann sums will over/under-approximate a definite integral. They view how the convergence of Riemann sums as the number of subintervals get larger.
Twelfth graders investigate applications of integration. In this Calculus lesson plan, 12th graders use the TI-89 to explore various problem solving techniques including symbolic, graphical, and numeric methods of solving applications of integrals.
In this calculus worksheet, students observe graphs and identify the limits of the functions listed in the graph. They determine the definite integrals and derivatives. Students use the trapezoid rule to estimate distance. This five-page worksheet contains 14 problems.
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an introductory course for anyone interested in mathematical structures with emphasis on computer implementation. The course includes topics such as propositional calculus, set theoretic concepts, relations and functions, mathematical induction, recursion, combinatorics, matrices, graphs, trees, their branching, leaves, and how to climb them (i.e. tree traversals). (3 credits)
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...
Show More learn problem-solving skills from the Bittinger hallmark five-step problem-solving process coupled with Connecting the Concepts and Aha! Exercises. As you have come to expect with any Bittinger text, we bring you a complete supplements package including MyMathLab™ and the New Instructor and Adjunct Support Manual. KEY TOPICS: Basics of Algebra and Graphing; Functions, Linear Equations, and Models; Systems of Linear Equations and Problem Solving; More Equations and Inequalities; Polynomials and Polynomial Functions; Rational Expressions, Equations, and Functions; Exponents and Radicals; Quadratic Functions and Equations; Exponential and Logarithmic Functions; Conic Sections; Sequences, Series, and the Binomial Theorem. MARKET: For all readers interested in Algebra
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Mathematical Excursions - 3rd edition
Summary: MATHEMATICAL EXCURSIONS, Third Edition, teaches students that mathematics is a system of knowing and understanding our surroundings. For example, sending information across the Internet is better understood when one understands prime numbers; the perils of radioactive waste take on new meaning when one understands exponential functions; and the efficiency of the flow of traffic through an intersection is more interesting after seeing the system of traffic lights represented in a math...show moreematical form. Students will learn those facets of mathematics that strengthen their quantitative understanding and expand the way they know, perceive, and comprehend their world. We hope you enjoy the journey102.89 +$3.99 s/h
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More About
This Book
Editorial Reviews
Booknews
A text for undergraduate students of mathematics, science, or engineering with a working knowledge of calculus, combining treatment of elementary theory of differential equations with material on methods of solution, analysis, and approximation. Features a large number and variety of problems, with emphasis on problems that call for conclusions to be drawn about the solution, and can be adapted to courses with varying levels of computer involvement. This sixth edition contains revised material on numerical methods, the method of Frobenius, and Fourier series, and an emphasis on visualization
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This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with...
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This classroom activity presents Calculus II students with some Flash tutorials involving work and pumping liquids along with some questions concerning the amount of work involved in pumping water out of two full containers having the same shape and size but different spatial orientations. Students are given opportunities to address this question by means of a write-pair-share activity in which they construct an integral equation and solve for an upper limit of integration.
An author's Snapshot for Mathematical Visualization Toolkit material found in MERLOT at...
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An author's Snapshot for Mathematical Visualization Toolkit material found in MERLOT at This material is a MERLOT classic award winner and an Editors choice winner. This snapshot shows an overview of the material. This was created in the MERLOT Content Builder.
After covering the standard course material on infinite series and their sums and the Integral Test for series convergence,...
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After covering the standard course material on infinite series and their sums and the Integral Test for series convergence, Calculus II students are given a write-pair-share activity that directs them to clearly explain the difference between a series and its related integral and explain why the sum of the series is greater than the value of the corresponding integral. Afterwards, the instructor employs a Web-based applet that visually displays graphs of both the series and the integral so that students can see the relationship between them.
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instructor was a visiting scholar and taught more theorems than other Linear Algebra sections. They are many methods in this course. Something like triangulization of matrices requires one to understand the math underlined
| 677.169 | 1 |
Change of Basis
A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.
Monday, August 25, 2014
Monday, August 18, 2014
Though I've been teaching at the college level for 15 years now, I've never been able to shake those first-day jitters. I have, however, gotten better at managing them and overcoming them and even having fun in the process.
This morning I taught my first Calc I class in...three years, I think? I believe this is the longest I've gone without teaching Calc I since I started teaching in grad school 15 years ago. That remove from the Calc I classroom, I believe, will help me come back to the subject with excitement and an authentic sense of novelty. I'm looking forward to a new format (for me), flipping the classroom and using computer-graded problem sets to give students practice in the basic computations they'll need to know.
Today we spent some time getting to know one another and reviewing some of the mathematical ideas leading up to calculus. I ended class by asking the students to write down (anonymously) any questions they might have about our class at this time...or if they had no questions, just to draw a smiley face.
Their questions, and my answers, are below, in no order other than the order I picked them up:
What's your biggest pet-peeve with your students? Disengagement. It really bugs me when students check out. Regardless of your skill level, give it some effort, please: we all have something we can give to the class, and we all have something we can take away.
Could we brush over some of the topics that were written on the board today, because even if I learned them, I need a little bit of remembering? Though we won't do much formal review in this class, we will do a lot of "just-in-time" review, meaning that as needed we'll brush up on important topics (e.g., trig identities, rules for exponents, factoring methods, etc.) when they come up. If you'd like more help, the Math Lab (located on the third floor of Robinson Hall) has many more resources and assistants who can help you, and you can always sign up with a private tutor. (You can find information about tutors from the Math Lab, too.)
I recently purchased a new Chromebook. Do you think all those softwares will be able to get on it? I'm assuming you're referring to Mathematica and LaTeX. I'm not sure about this, but it's a great question. I believe if you run a flavor of Linux as your OS on the Chromebook you should be able to find both of these applications by visiting the websites I've linked to on the class website. If you run the standard Chrome OS, though, I'm not sure. I'll look into this.
If we want to practice/review some old (algebra or similar) concepts, is there a site for problems? Good question. I believe the Math Lab staff can point you in the direction of some good texts and e-resources for review. Visit them!
Are there any other calc tutors other than the Math Lab? You can sign up with a private tutor, a list of whom is available by visiting the Math Lab.
Explain what the "lab" section of the class is. For arcane bureaucratic reasons, the department must differentiate between the "standard" class and the "lab," but besides the difference in time and place, there's no other distinction. We'll do many of the same things on all days. Just show up, relax, and have fun!
Can I eat breakfast in class? Yup. No worries! Just try not to spill anything or eat too loudly.
What is the meaning of life? Dunno. I missed that day.
:)
Q: What is beauty? What is truth? A: Type "graph " into Wolfram Alpha. I'll get on this at once and report back.
Ready for Calc! :)
How many chapters will be covered? We'll discuss the information addressed by the first five chapters of the textbook. I should point out, though, that I despise the term "cover" in reference to education. It corresponds to a very antiquated notion of pedagogy, in which the teacher "covers" material by talking about it in front of the students, thereby absolving the teacher of all guilt in case the students do poorly on an exam. "I can't believe they did so poorly on this question...we covered it in class!"
Is most of the reading done in class or not? Not. You will be responsible for reading the textbook on your own time. I would guess that a somewhat careful reading of most sections should take you about an hour, and we'll discuss roughly two or three sections per week. You do the math!
:-)
Can we review trigonometry? Pre-cal was a long time ago... See my comment above about "just-in-time" review. We'll definitely review trig and other tricky concepts as needed, when needed.
Wednesday, January 15, 2014
Halfway through our winter break I paused in my course
preparations, panicking that I'd spent nearly all of my time planning my
HON 479 course and had done nearly nothing to prep for Linear Algebra II.
Though I had a rough framework for the course's structure (semiregular
homework assignments, a few take-home exams, student-led projects,
presentations, and discussions), I had almost no idea what content I would include in the course.
After a few moments (okay, maybe a few hours), the panic passed. I realized the futility of overplanning, a futility reconfirmed by the survey of my Linear II students' background I performed on Monday. The 23 students in that class come to me having taken Linear Algebra I from no fewer than five different faculty members in my department, as long ago as two and a half years back. These faculty include me and one of my colleagues who shares my penchant for student-centered, application-based teaching, a couple folks who typically offer a blend of applications and theory (one with a much more student-centered approach than the other), and a fifth who focuses exclusively on abstraction and theory and whose teaching style can only be described as "traditional." Needless to say, my 23 students come to me with extremely diverse linear algebraic backgrounds. It's unlikely that, beyond a few basic principles (row reduction, linear (in)dependence, bases, determinants, eigenvalues and -vectors, etc.) they all will have studied, they'll have any content knowledge in common. In the end there's really very little I can do to accommodate them all: no matter what static plan for the course that I could come up with, it would no doubt lose some and bore most of the others.
This realization was liberating. Instead of putting forth a particular course of study, I could let the students take the lead, offering them the chance to investigate topics in which they are interested, sharing their investigations with each other in the form of in-class presentations, discussions, and problem sets. I'm going to ask every student to take a turn, working with one or two of her or his peers, leading the class in the study of a topic of her or his choosing. For those who might not know what direction they'd like to head in, I made a list of potential topics, many of which likely made an appearance in some students' first-semester Linear I courses:
To help everyone get to the point where we can approach some of these topics, I'm spending the first week or two on review, wherein the students are taking turns, in groups of three or four, presenting on the various "basic principles" I listed above. It's going well so far. "Is this useful at all?" I asked after a couple of presentations this morning. "Should we keep doing this?" There was almost unanimous agreement that yes, we should. So we'll keep it up.
Tuesday, January 14, 2014
The Honors Program has become a victim of its own success, in a way: so many students are now pursuing Distinction as a University Scholar that I'm finding it necessary to offer not one but two sections of HON 479 this term...and perhaps in (nearly?) every term for the indefinite future. Of course, by the time that I realized the need for a second section of the course (about halfway through last semester's advising period), it was really too late to find another instructor to teach that section, and besides, I'd prefer to have a single instructor for both sections, for consistency's sake. Of course, that meant that the instructor for the first section would also be the instructor for the second section, even if it meant (as it did) teaching a course over the normal load. Of course, that instructor is me.
Genius that I am, I scheduled the two sections to meet back-to-back, 100 minutes apiece with only ten minutes in between, every Tuesday and Thursday. Today was our first meeting. By the end of the first section my throat hurt, and by the end of the second I had nearly no voice: there are so many moving parts to this class that I've just got to spend much of the first class meeting pointing out just how all of those parts fit together and more in a meaningful way.
Throw in a minor student medical emergency, a pressing tech issue facing the school's student-run TEDx chapter (for which I'm the faculty adviser), various administrivia and bureaucratic bullshit, and a two-hour sojourn in Asheville Catholic School's gymnasium, where I helped a friend out as a middle-school science fair judge, and you've got a hell of a day. I'm sore-throated and brain-dead, and I'm tired as hell.
But I'm happy. I've got high hopes for this term. I feel like last semester gave me a good grip on 479, and I had a fantastic first meeting of Linear Algebra II yesterday (the 23 students in that class had five different instructors for Linear I!), a course which I'll be teaching from a nearly total project-based perspective.
Tuesday, December 10, 2013
It's the last day of final exams, and students are trickling in to say their "farewells" and their "see ya next years." Some are saying yet more long-lasting goodbyes, graduation soon to take them far away.
An hour or so ago three of our more outgoing Honors students came by to bid adieu to Queshia and me. They sat in the Honors office and we talked for about twenty minutes. Much of our conversation centered on the idea of letting go...or not: end-of-semester goodbyes, helicopter parenting, and relationships that have run their course.
"Kids these days," I began, noting that I'd once sworn up and down that I'd never say those words, "don't have the freedom we had when I was young. As long as my parents knew roughly where I was, we were free to roam about the town, with little worry what we'd get into." Now, of course, parents hover overhead. They call the program on their kids' behalves, inquiring about requirements and expectations and perks. They ask after every detail of their kids' academic lives. They have to learn to let go.
We all have to learn to do this, and it's not an easy thing to do.
As I wrote in a recent post, my life lately has been filled with loved ones lost. One of my closest friends lost her mother, suddenly, and not two weeks later another friend, just as suddenly, lost her father. In the skinny interstice between these deaths two other, yet younger, friends of friends passed away, and in the time since my friend's father's death I've heard several talk of losing parents, friends, and pets. It's gotten overwhelming, and, as I hinted in that same post above, I can't say that I've handled it well.
Why not? One reason, I think, is that I've been lucky enough to not have lost many people truly close to me. I've made it through 38 years without losing a particularly close friend or family member. I seem to be blessed with a particularly healthy set of childhood friends, and my friends from college are no less hearty and robust. And my mom and dad both up and moved far, far away from their respective families when they were young, so I grew up hell and gone from my extended families. This meant that I hardly knew any of the grandparents and other more distant relatives I've lost, having only seen them for a few days at a time once every other year or so, and then only when I was very young. We simply weren't close.
I don't mean to sound unfeeling or callous or cold: this is just the way it is. I've never had to deal directly with death; I'm as yet unfamiliar with its effects on me. What's more, I still don't feel as though I'm dealing with it directly, even now, but really only through others, and thus I'm not so much dealing with death as I am dealing with the effects that death has on my relationships with those dealing with death directly. Therefore my experience is a mediated one and, because it centers on others' relationships with me, it's an experience I thought at first was necessarily selfish.
But does it have to be selfish? On reflection, I think not.
When tragedy strikes our friends, we can choose to remove ourselves and feel their pain only through the effects it has on the relationships we share with those friends. We see the tragedy strike, but we don't feel it immediately. We shelter ourselves. We may offer our support, but that support is academic, it's detached and distant.
I fear that this is the kind of support I've been offering to my friends in their recent mourning. I've baked a few dozen cookies and a couple loaves of bread, I've offered the expected words of solace and succor, and I've offered a hand with transportation and child care, if needed. But I've not really been present for the pain. I've spent more time focusing on the way in which the various tragedies affect me, as mediated through my friends' pain in turn.
I need to learn to let go.
I need to learn to let go of my own pain, to feel it, but also to let it pass so that in its place I can place a picture of the pain my friends may be going through as they deal with their loss.
Further, I need to learn to let go of my self, if only for a little while, to see beyond my self and my immediate relationships with my friends, to see instead to my friends' relationships with the loved ones they've lost.
Finally, I need to learn to let go of those very relationships, or at least my static conceptions of those relationships, and to accept that tragedy brings great waves of change and that once those waves have passed the relationships they've left behind might look very different than they did just days before.
To anyone to whom I've not been able to offer the succor or support you've needed from me, I apologize. I've not before dealt with death so directly, and I'm only now learning my own authentic reactions to it. I'm a work in progress, and that progress may be slow at times, but I promise you that it's there.
Sunday, December 08, 2013
I'm now three semesters into my stint as Honors Program director, and I think I'm starting to get the hang of the gig. I've learned the ropes well enough to feel confident tweaking things here, cinching it up there, and making many many midcourse adjustments. Give me another term or two and I'm gonna feel ready to make some bigger changes.
Like what? I've had a number of conversations with one of my closest colleagues about ways in which the Honors Program could be made to cater more to students who demonstrate exceptional intellect and motivation via measures other than standardized test scores and high achievement in courses (like AP classes) ultimately driven by rote examination. I don't want to go too far too fast, but my colleague and I brainstormed ways we could modify both the admission process to the program and the requirements for graduation with Distinction as a University Scholar in order to encourage less the grinds, grade-grubbers, and résumé-builders (many of whom either drop from the program before completing Distinction requirements or simply take a path of least resistance, relying on courses they know won't really challenge them) and more the risk-takers, visionaries, and authentic learners (many of whom are ineligible for the program as it's currently constructed because their risk-taking and earnest focus on real learning has led them to lower performance by quantitative measures).
How might we do this? Disallow membership in the program for first-year (or at least first-semester) students, requiring all interested students to opt in to the program (and not simply be placed there) after having spent some time at the university. Admittance criteria would be more holistic and not so focused on classroom performance. The program's curricular offerings would be more intentionally integrative and dovetail with substantial extra- and co-curricular activities and programming. Students would be asked to complete a sort of Honors thesis at the end of their involvement in the program. Most important, Honors students would be asked to interact in a meaningful fashion with students who are not members of the program. Of what this interaction would consist...I don't know. All I know now is that both I and my partner in crime in this revisioning exercise believe that the Honors Program offers a troubling equity issue, providing real resources to the most academically gifted of students, the ones who are less likely to need those resources in order to succeed in their college careers, while their less-academically-gifted peers make do without such assistance.
Excellence without elitism: how do we realize this vision? One way might be to take the tack we've slowly been turning to over the last couple of terms, emphasizing not the Honors Program's academic offerings but instead its sense of community. I truly believe we've done far more to support Honors students' success during the past year through Honors yoga sessions, Reading-Day snacks, "Good Books" reading groups, and Honors trivia nights than we have through sending a small handful of Honors students to statewide, regional, and national conferences.
My university (like every other in the country) is struggling with recruitment and retention, and I truly believe the community-building we're trying to do in the Honors Program is an unbeatable means of achieving those two related goals. Nothing beats the inestimable and intangible benefit of bringing the
students together in the Laurel Forum, introducing those with like
interests and aims, giving them access to one another's support. They'll stick around, and they'll not regret a minute of it. And when their younger peers come to visit the school they'll talk the program up into the stratosphere (I've heard them do it).
So, expect to see more community-building as the program looks to the future. And if you've got any ideas for ways we can do this (jigsaw puzzles? Brew-offs? Iron-Chef-like cooking competitions?), please let me know.
Friday, December 06, 2013
Yesterday Nelson Mandela died, and the world lost one of its greatest ever agents of peace. Meanwhile, more locally, during the past few weeks several people very close to me have dealt with the deaths of too many loved ones to count: a father, a mother, and so, so many friends (one covered with once-soft black fur). It's been a very rough month, and I don't believe I've handled it as well as I might have. I don't think I've been as present as I could have been; I think I've been too self-absorbed. I've been sleepwalking, but I feel as though I'm coming awake.
Yesterday my HON 479 students put on their long-awaited workshop on
diversity, inclusion, and equity, focusing on the ways in which these
manifest in religion, race, and gender. They worked with a small audience comprising about ten faculty and staff and a couple of their fellow students. The group was small, but it was engaged. The conversations we had were rich, heartfelt, authentic. The event was enlightening, meaningful, and moving. Working with wonderful visuals (Like the Cooper Center's "Racial Dot Map" and It's Pronounced Metrosexual's "Genderbread Person v. 2.0") and excellent activities ("The Cold Wind Blows," religious insensitivity role-plays, and a few rounds of reflective writing on our own gender and racial identities), the students' workshop was substantially better than the awful diversity and inclusion workshop I took part in earlier this year.
At some point late in yesterday's workshop's proceedings, while one of my students was talking about her intellectual journey as a devout Christian completing an academic degree in religious studies, I had a sudden feeling of self-awareness. It was a feeling of being and becoming all at once. It was also a feeling of oneness, of unity with the people I'd just shared the past two hours with. Though our group was small, we represented several races and ethnicities, several religious traditions, several gender and sexual identities. We offered a substantial cross-section of our society, and we were having civil...nay, collegial, even cordial...conversation on some of the most difficult topics for anyone to talk about.
On the way home from campus, NPR told me that Mandela had died, and I teared up in the car. I thought of South Africa's Madiba, and his friends and colleagues in struggle. I thought in particular of Archbishop Desmond Tutu, for whom I have more respect than nearly anyone else. In his incredible book No Future Without Forgiveness Tutu speaks of the concept of ubuntu, a Bantu term referring to our human interconnectedness, which Mandela once described as follows:
"A traveller through a country would stop at a village and he didn't have
to ask for food or for water. Once he stops, the people give him food
and attend him. That is one aspect of Ubuntu, but it will have various
aspects. Ubuntu does not mean that people should not enrich themselves.
The question therefore is: Are you going to do so in order to enable the
community around you to be able to improve?"
In the short time I had before I had to leave again to meet with my writing group I sliced onions for the simple meal of lentils and rice I would make when I came home again, and I read the first lines of the end-of-semester reflection one of my HON 479 students had handed to me just before helping to host that afternoon's workshop. "Dear Patrick, I hope you don't find this format too informal," it begins. "I tailored my response with you in mind, so I thought I might address you directly." In tandem with her humble letter (which brought me to tears by the time I was done reading it) was a hand-made jigsaw puzzle the student had crafted.
I had no time to assemble more than the frame of the puzzle before leaving, but completing the puzzle was the second thing (after starting dinner) I did on my return home.
"You may start to notice (or maybe you have finished) that the puzzle is a tree. I chose a tree because I think it represents various aspects of the IHAD program."
"Now, I'm sorry to deprive you of the satisfaction of putting that last piece in the puzzle, but I did not lose it and neither did you. How frustrating is it to complete a process yet still feel as though you are missing something?...Thank you for going through this puzzling process with me today."
I've recently taken to origami, more seriously than my halfhearted efforts in the past. I am struck in particular by the beauty and meaning of the kusudama, or "medicine ball," a form that's meant to ward off evil and encourage health and strength. I made a kusudama a week ago for a grieving loved one, and I'm making more now, for friends, for family, for people I love. I fear I'll never stop, for right now I feel a sort of universal love which I hope I'll never lose.
It's a new day. As this day begins, please take a moment to love yourself, to love each other, to find peace and joy in all that you do. Enjoy being, but keep becoming.
| 677.169 | 1 |
Precalculus
9780073312637
ISBN:
0073312630
Edition: 6 Pub Date: 2007 Publisher: McGraw-Hill College
Summary: The Barnett, Ziegler, Byleen College Algebra series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. Precalculus introduces a unit circle approach to trigonometry and can be used in one or two semester college algebra with trig or precalculus courses. The large number of peda...gogical devices employed in this text will guide a student through the course. Integrated throughout the text, students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD reinforces important concepts, and provides students with extra practice problems.
Barnett, Raymond A. is the author of Precalculus, published 2007 under ISBN 9780073312637 and 0073312630. Twenty two Precalculus textbooks are available for sale on ValoreBooks.com, ten used from the cheapest price of $60.70, or buy new starting at $120.14
| 677.169 | 1 |
Description of Item
Alpha Omega Publications makes learning about statistics and graphs easy and fun with the LIFEPAC Pre-Algebra & Pre-Geometry I Unit 9 Worktext. This ninth in a series of ten worktexts covers the gathering and organizing of data, central tendency and dispersion, graphs of statistics, and graphs of points. With the help of this outstanding homeschool math curriculum, your seventh grader will be able explain statistics like a pro!
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The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the...
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Math Algorithms
Description
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Course Content and Outcome Guide for ALC 95D
Course Number:
ALC 95D
Course Title:
Math 95 Lab - 3 credits
Credit Hours:
3
Lecture Hours:
0
Lecture/Lab Hours:
0
Lab Hours:
90
Special Fee:
$36.00
Course Description
Provides a review of individually chosen topics in Intermediate Algebra (Math 95). Requires a minimum of 90 hours in the lab. Completion of this course does not meet prerequisite requirements for other math courses.
Intended Outcomes for the course
Upon successful completion of this course students will be able to:
Choose and perform accurate intermediate-level algebraic computations in a variety of situations with and without a calculator.
Solve a problem at home or in an academic or work environment by creating an intermediate-level algebraic expression or equation that represents the situation and find the solution to the problem using correct intermediate-level algebraic steps.
Recognize patterns in data collected or observed at home or in an academic or work environment and use the observed patterns to make predictions.
Outcome Assessment Strategies
Assessment shall include at least two of the following measures: 1. Tests 2. Attendance 3. Portfolios 4. Individual student conference
Course Content (Themes, Concepts, Issues and Skills)
Intermediate Algebra (MTH 95) Themes: Functions and function notation Functions represented as graphs, tables, equations and in words Connection between symbolic and graphical representations Algebraic simplification of expressions and solving of equations Problem solving and modeling, interpreting results in practical terms Language of graphs Skills: Calculator (integrated throughout the course) Use the home screen carry out arithmetic operations Use the calculator's table feature to explore functions Graph functions . Input the appropriate window settings to view the graph . Use calculation tools Value Zero Maximum Minimum ntersect Understand that the calculator has limitations Functions Understand and apply the definition of function Determine whether one quantity is a function of another algebraically, graphically, numerically and within real life contexts by applying the definition of a function Domain . Understand the definition of domain (the set of all possible inputs) Determine the domain of functions represented graphically, algebraically, numerically and verbally Represent the domain in both interval and set notation, where appropriate Apply unions and intersections (€œand€ and €œor€) when finding and stating the domain of functions Understand how the context of a function used as a model can limit the domain Range . Understand the definition of range (set of all possible outputs) . Determine the range of functions represented graphically, numerically and verbally . Represent the range in interval and set notation, where appropriate Function notation Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically, numerically and verbally (e.g. evaluate ) Algebraically simplify and distinguish between different examples such as , , and nterpret in the appropriate context e.g. interpret where models a real-world function Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e. solve for and solve for where and should include but not be limited to linear functions, quadratic functions, and absolute value functions) Solve function inequalities algebraically (i.e. , , and where and are linear functions and and where is an absolute value function) Solve function inequalities graphically (i.e. , , and where and should include but not be limited to linear functions, and for quadratic and absolute value functions) Graphs of functions Use the language of graphs and understand how to present answers to questions based on the graph (i.e. read the value of an intersection to solve an equation and understand that is a number not a point) Determine function values, solve equations and inequalities, and find domain and range given a graph Apply function notation to prerequisite skill of finding linear equations given two ordered pairs Quadratics Recognize a quadratic equation given in standard form, vertex form and factored form Solve quadratic equations by completing the square Find complex solutions to quadratic equations by the quadratic formula or by completing the square Understand the graphical implications (i.e. when there is a complex number as a solution to a quadratic equation) nterpret the meaning in the context of an application Quadratic functions in vertex form Graph a parabola after obtaining the vertex form of the equation by completing the square Given a quadratic function in vertex form or as a graph, observe the vertical shift and horizontal shift of the graph Connect graphing via vertex form with the prerequisite graphing methods (i.e. axis of symmetry, horizontal intercepts, vertex formula, vertical intercept, points found by symmetry) Determine the domain and range of quadratic functions algebraically and graphically Applications Understanding in context: given a quadratic function in algebraic or graphical form find and interpret, including units, the meaning of the: Vertex as a maximum or minimum Vertical intercept Zeroes/horizontal intercepts/roots nputs and outputs of functions (e.g. and ) Clearly define variables including appropriate units State conclusions to applied problems in complete sentences including appropriate units Explore quadratic functions graphically using the graphing calculator. Convey results using function notation. Examine the following features: . Vertex . Vertical intercept . Horizontal intercepts Radical Functions Understand nth roots Determine the domain of radical functions with both even and odd roots algebraically and graphically Determine the range graphically Understand radicals as expressions with rational exponents and vice versa Use rational exponents to simplify radical expressions (See addendum) Practice prerequisite skills of exponents rules in the context of rational exponents Rationalize denominators so students can recognize equivalent expressions (e.g. ) Solve radical equations algebraically and graphically . Verify solutions algebraically Understand that extraneous solutions found algebraically do not appear as solutions on the graph . Solve literal radical equations for a specified variable Calculator . Approximate radicals as powers with rational exponents . Find the domain and range of radical functions . Solve radical equations graphically . Use graphical solutions to check the validity of algebraic solutions Rational Functions Determine the domain of rational functions algebraically and graphically Simplify rational functions, understanding that domain conditions lost during simplification MUST be noted Rewrite rational expressions by . Canceling factors common to the numerator and denominator . Multiplying . Dividing using both and notation Simplify the following cases where a, b, c, d represent real numbers, linear polynomials or quadratic polynomials: , and . (See addendum) . Adding . Subtracting . Simplifying complex rational expressions The following forms of complex rational expressions shall be simplified: , , , and where , , , , , and represent real numbers, linear polynomials in one variable, or quadratic polynomials in one variable. (See addendum.) Solve rational equations . Check solutions algebraically Solve literal rational equations for a specified variable . Introduce variables with subscripts Applications . Solve distance, rate and time problems involving rational terms using well defined variables and stating conclusions in complete sentences including appropriate units . Solve problems involving work rates using well defined variables and stating conclusions in complete sentences including appropriate units Addendum Functions should be studied symbolically, graphically, numerically and verbally. As much as possible, instructors should present functions that model real-world problems and relationships to address the content outlined on this CCOG. Function notation is emphasized and should be used whenever it is appropriate in the course. Students should be required to use proper mathematical language and notation. This includes using equal signs appropriately, labeling and scaling the axes of graphs appropriately, using correct units throughout the problem solving process, conveying answers in complete sentences when appropriate, and in general, using the required symbols correctly. Students should understand the fundamental differences between expressions and equations including their definitions and proper notations. All mathematical work should be organized so that it is clear and obvious what techniques the student employed to find his answer. Showing scratch work in the middle of a problem is not acceptable. Since technology is used throughout the course, there is a required calculator packet for students that gives directions for several graphing calculators. The students should understand the limitations of calculator€"i.e. when the calculator gives misleading information. Examples of the calculator€™s limitations include the following: when finding horizontal intercepts, the calculator sometimes gives something like y = 3E-13; the calculator rounds to 12 or fewer decimal places; some calculators appear to show vertical asymptotes on the graphs of rational functions; it appears that the graph of touches the x axis; the calculator does not show holes on rational function graphs; the calculator cannot handle very large numbers, e.g. etc. For dividing rational expressions as in 5.3.3 and 5.3.3.1, focus on examples where the letters represent real numbers and linear polynomials. E.g. , , and .Exploration of difficult rational exponents, as in 4.5, should be limited. Basic understanding is essential and a deep understanding takes more than one course to develop. Examples should be limited to one or two variables, keeping things as simple as possible while covering all possibilities. E.g. , , , . As much as possible, instructors should present functions that model real-world problems and relationships to address the content outlined on this CCOG. In 3.3.1, when solving applications of quadratic equations, a complex solution should be interpreted as the graph never reaching a particular real world y-value. For simplifying complex rational expressions as in 5.3.6.1, a major emphasis shall be placed on cases where , , , , , and (as above) represent real numbers, linear polynomials in one variable. For example, or would be good examples.
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Some mathematical skills are essential for engineering and science courses. Mostly it will be assumed that these skills have already been mastered, and unless so, it is easy to become lost in further study. This study guide is intended to help drum in these skills and keep them there.
The theory of homogenization replaces a real composite material with an imaginary homogeneous one, to describe the macroscopic properties of the composite using the properties of the microscopic structure. This work illustrates the relevant mathematics, logic and methodology with examples.
The second in this two-volume series also contains original papers commissioned from prominent 20th-century mathematicians. A three-part treatment covers mathematical methods, statistical and scheduling studies, and physical phenomena. 1961 edition.
Provides a range of worked mathematical examples which are appropriate for scientists and engineers, ranging from basic algebra to calculus and Fourier transforms. This book also summarizes the basic concepts and results and covers the material needed by science undergraduates
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Elementary Algebra-Student's Solutions Manual - 8th edition
Summary: When the answer at the back of the book is simply not enough, then you need the Student Solutions Manual. With fully worked-out solutions to all odd-numbered text problems, the Student Solutions Manual lets you "learn by example" and see the mathematical steps required to reach a solution. Worked-out problems included in the Solutions Manual are carefully selected from the textbook as representative of each section's exercise sets so you can follow-along ...show moreand study more effectively. The Student Solutions Manual is simply the fastest way to see your mistakes, improve learning, and get better gradesellBackYourBook Aurora, IL
032156733136 +$3.99 s/h
Acceptable
Barnacle Stanton, CA
2009-03-09 Paperback Fair Acceptable. Soiled. From California94
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Product Details
See What's Inside
Product Description
The distributive principle is one of the most important and widely used concepts in mathematics. Though interlaced with other mathematical concepts, it provides a spotlight on major educational and mathematical themes. It is a significant tool in seeking ways of relating algebraic and geometric thinking and of expanding their very different potential to formulate generalizations. Stephen Brown, through his research, writings, and practice in the classroom, has discovered the deep and powerful educational and mathematical implications of applying the distributive principle as a way to encourage students to relate mathematics to the "real world," to seek connections beyond science and mathematics, and to include literature and language as ways of expanding their mathematical thinking.
This extraordinary collection of essays, each of which uses the distributive principle to shed light on a different aspect of mathematical thought, encourages readers to find new ways of understanding the significance and limitations of a subject that is usually presented as purely logical and not often connected with how the mind works in other domains. Using the distributive property as a vantage point, the essays cast a wide net. For example, in one essay the connections between algebraic and geometric variations of the property are developed in the context of presenting enticing problems. Though prime numbers are part of the curriculum (in such diverse topics as the infinitude of primes, the unique factorization of composites into a product of primes, and the ability to reduce all fractions to "lowest terms"), in focusing on the distributive property, the author expands on the concept of prime numbers in ways that point out the deep significance of concepts that are often prematurely taken for granted.Related ProductsNovember 2013 Focus Issue: Beginning Algebra: Teaching Key Concepts
How can all students learn the key concepts of beginning algebra? Through the study of algebra, students learn to think abstractly, apply various representations, communicate mathematically, and develop the habits of mind that are needed to use mathematics and become lifelong learners. Whether taught within a first-year algebra or an integrated course, algebraic concepts form a core of mathematical knowledge that students need for future success. The Editorial Panel of Mathematics Teacher solicits manuscripts that examine ways to teach the key mathematical concepts students must learn in a beginning algebra course.
Mathematical reasoning begins with questions rather than answers. The 2011 Focus Issue highlights the innovators who foster children's natural curiosity as a key mechanism for teaching and learning mathThe ideas and advice from experienced educators can help beginning middle school teachers create an environment in which students view mathematics as useful and exciting and become prepared for more in-depth study of mathematics in high school and beyond.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Mathematics in Our World
"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster ...Show synopsis"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster interest in and show the applicability of mathematics. The book is written by our successful statistics author, Allan Bluman. His easy-going writing style and step-by-step approach make this text very readable and accessible to lower-level students. The text contains many pedagogical features designed to both aid the student and instill a sense that mathematics is not just adding and subtracting Mathematics in Our World with MathZone. This book is in...Good. Mathematics in Our World with MathZone. This book is in Good condition. Buy with confidence. We ship from multiple location.
Description:Good. 0073311820 USED BOOK in good condition| No supplements|...Good. 0073311820 USED BOOK in good condition| No supplements| Normal wear to cover, edges, spine, corners, and pages | Writing / highlighting | Inventory stickers | Satisfaction guaranteed!
Description:New. 0073311820 AtAGlance Books--Orders ship next business day,...New. 0073311820 AtAGlance Books--Orders ship next business day, with tracking numbers, from our warehouse in upstate NY. This book is in brand new condition.
Reviews of Mathematics in Our World
This book was in terrific condition and came sooner than expected for my husband's college course. Even better, the book is the teacher's edition and had some great helps for those of you who are rusty on math procedures for different real-world applications.
If you need tyo brush up your math skills in any way, this is the book for it, especially if you get the teacher's
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Use Wolfram|Alpha to Solve Calculus Problems and…...
Use Wolfram|Alpha to Solve Calculus Problems and… Everything Else.
Wolfram|Alpha is like Google on crack. However, it is not technically a search engine; it is a "computational knowledge" engine. They use a huge collection of trustworthy, built-in data to get the user the information or knowledge they are looking for. When you search for an item, Wolfram|Alpha gives you all of the relevant knowledge they have on that specific search query. For example, here is the results for the search "when did the Beatles break up?" Not only do you get the date the Beatles broke up, you also get how long away that date is from today and other noteworthy events that occurred on the same day. Here is another example, for the search "carbon footprint driving 536 miles at 32mpg" that tells you the amount of fuel consumed and the amount of c02 and carbon emitted.
Because Wolfram|Alpha is just retrieving answers from its huge database of information and formulas, you have to be specific and ask non-opinionated questions. For example, the website does not know which Lil Wayne song is the best. However, it does know things that are not opinions, like the nutritional facts of 10,000 big macs and how many planes are currently flying directly over you.
I find Wolfram|Alpha to be better than Google when I am quickly looking for specific answers. I just typed in "Countries that border France" on both Wolfram|Alpha and Google. Wolfram|Alpha quickly showed me a list of the 8 countries and a map with of France with its bordering countries highlighted. Google on the other hand sent me over to Yahoo Answers…
Other than a fun search engine, Wolfram|Alpha can also be used as a highly effective tool for college. Like the title mentions, the knowledge engine can in fact solve any calculus problem. It can easily solve any math problem thrown its way, from a basic algebra problem to whatever this is.
Wolfram|Alpha can also be used for many other college courses such as biology, astronomy, history, etc.
As Wolfram|Alpha can be kind of confusing and hard to get the hang of at first, I suggest going through this short tour and looking at some examples to help give you a better sense of how to use it. Even if you find it a little bit confusing at first, keep trying because Wolfram|Alpha really is a great way to "hack college."
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Big Ideas Learning Releases New Common Core High School Math Series
Big Ideas Learning, LLC is pleased to introduce the latest addition to its Common Core Math Curriculum: The Big Ideas Math Algebra 1, Geometry, and Algebra 2 High School series. This series was written by renowned authors Dr. Ron Larson and Dr. Laurie Boswell, the same team that produced the industry-leading Big Ideas Math Middle School series.
The Big Ideas Math 2015 Common Core High School series was designed to challenge students beyond basic memorization and to provide them with all of the resources they need to apply mathematical concepts to real-life scenarios and to questions found on standardized tests. The series was written to the Common Core State Standards and uses the Standards for Mathematical Practice as its foundation.
An innovative technology package has also been interwoven into the Big Ideas Math High School series. The Big Ideas Math Dynamic Assessment and Progress Monitoring Tool with adaptive functionality, Interactive eBook App, and Interactive Whiteboard Lesson Library are only a few of the brand-new resources available for students and educators.
"It is our goal to provide students and educators with everything they need to create the most effective learning environment possible," said Denise McDowell, Vice President of Sales, Marketing, and Curriculum at Big Ideas Learning. "The Big Ideas Math 2015 Common Core High School series fully equips students with the resources they need to be successful in mathematics, and fully equips teachers with the tools necessary to know where their students are and monitor student progress as they guide their students to where they need to be."
About Big Ideas Learning, LLC Big Ideas Learning, LLC is a wholly owned subsidiary of Larson Texts, Inc. Big Ideas Learning, LLC and Larson Texts, Inc. produce student-friendly, market-leading math textbooks for sixth grade through college calculus that are used by over five million students each year. Larson Texts has been deeply committed to providing innovative and coherent print and online materials to the education community for more than 30 years. The textbooks produced by Big Ideas Learning, LLC and Larson Texts, Inc. have become distinguished for their readability, accuracy, and real-life applications. For more information, please visit
For more information about Big Ideas Learning or to schedule an interview with Ron Larson, please e-mail Lauren Stefanick at lstefanick@larsontexts.com or call her at (814) 824-6365.
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This intermediate algebra text was developed in the spirit of the AMATYC standards for problem-solving, modeling, connecting with other disciplines, technology, and calculus reform. Many traditional topics are included in the text, but the development and application of these topics is not traditional. This book allow students to take an active role in the learning process through collaborative learning techniques, hands-on activities, and use of the graphing calculator as a problem-solving tool. Geometric examples and problems are integrated throughout the text. Students explore topics through various activities and use calculator based laboratory (CBL) experiments to generate data describing real-world situations and applications.
Book Description:Addison-Wesley. Book Condition: New. Original Books from across the World in Brand New Condition --------- It can be shipped from U.S, Canada, Germany & India/Overseas centre subject to availability. 9780201853629. book. Bookseller Inventory # RM_DB_L-5885
Book Description:Pearson. PAPERBACK. Book Condition: New. 0201853620201853620ZN
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Algebra and Trigonometry
Mike Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing ...Show synopsisMike Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. In the Ninth Edition , Algebra and Trigonometry has evolved to meet today??? s course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or onlineThe previous version of this textbook is better than this version. The examples are not as defined as the previous version, I think that is important in a book. In buying this textbook, I recommend one transfers to another classroom that uses anohter
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Discrete Mathematics for Teachers Preliminary Edition
9780618433926
ISBN:
0618433929
Edition: 1 Pub Date: 2004 Publisher: Houghton Mifflin College Div
Summary: Part of a new generation of textbooks for in-service and pre-service teachers at the junior-senior level, this text teaches in three main ways: it extends students' breadth of knowledge beyond, but related to, the topics covered in elementary and middle-grade curriculums; it increases prospective teachers' depth of mathematical understanding by providing problems rich in exploration and mathematical communication; an...d it models the most current ways of teaching mathematics. Many Section Openers begin with a motivating lesson that introduces a new topic in an understandable, real-world context. Exercise Sets at the end of every section provide more traditional practice and are labeled either Proof Exercises or Writing Exercises. Exploratory Exercises at the end of every section lead students to investigate topics outside the framework presented in the section. The final exercise in each section is a writing exercise.
Wheeler, Ed is the author of Discrete Mathematics for Teachers Preliminary Edition, published 2004 under ISBN 9780618433926 and 0618433929. Twenty three Discrete Mathematics for Teachers Preliminary Edition textbooks are available for sale on ValoreBooks.com, twelve used from the cheapest price of $57.25, or buy new starting at $108.00
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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ISBN: 0618394370 / ISBN-13: 9780618394371
College Algebra: A Graphing Approach
Part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, COLLEGE ALGEBRA: A GRAPHING APPROACH, 5/e, is an ideal user ...Show synopsisPart of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, COLLEGE ALGEBRA: A GRAPHING APPROACH, 5/e, is an ideal user resource for courses that require the use of a graphing calculator. The quality and quantity of the exercises, combined with interesting applications and innovative resources, make teaching easier and help users succeed. Continuing the series' emphasis on user support, the Fifth Edition introduces Prerequisite Skills Review. For selected examples throughout the book, the Prerequisite Skills Review directs users to previous sections in the book to review concepts and skills needed to master the material at hand. In addition, prerequisite skills review exercises in Eduspace (see below for description) are referenced in every exercise set. The Larson team achieves accessibility through careful writing and design, including examples with detailed solutions that begin and end on the same page, which maximizes the readability of the book. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. This Enhanced Edition includes instant access to Enhanced WebAssignA(R), the most widely-used and reliable homework system. Enhanced WebAssignA(R) presents thousands of problems, links to relevant book sections, video examples, problem-specific tutorials, and more, that help users grasp the concepts needed to succeed in this course. As an added bonus, the Start Smart Guide has been bound into this book. This guide contains instructions to help users learn the basics of WebAssign quickly.Hide synopsis
...Show more includes a moderate review of algebra to help students entering the course with weak algebra skills.Hide
Description:Good. 0618394370 Textbook in good condition. Pages are clean...Good. 0618394370 Textbook in good condition. Pages are clean and tight. Covers have some wear and creasing
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Mathematical reflections : in a room with many mirrors by Peter Hilton(
Book
) 18
editions published
between
1996
and
1998
in
English and Italian
and held by
719 WorldCat member
libraries
worldwide
The purpose of this book is to show what mathematics is about, how it is done, and what it is good for. The relaxed and informal presentation conveys the joy of mathematical discovery and insight and makes it clear that mathematics can be an exciting and engrossing activity. Frequent questions lead the reader to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful ideas. The text presents eight topics that serve to illustrate the unity of mathematical thought as well as the diversity of mathematical ideas. Drawn from both "pure" and "applied" mathematics, they include: spirals in nature and in mathematics; the modern topic of fractals and the ancient topic of Fibonacci numbers; Pascal's Triangle and paper folding -- two topics where geometry, number theory, and algebra meet and interact; modular arithmetic and the arithmetic of the infinite. The final chapter presents some ideas about how mathematics should be done, and hence, how it should be taught; these ideas are referred to throughout the text, whenever mathematical strategy and technique are at issue. Presenting many recent discoveries that lead to interesting open questions, the book can serve as the main text in courses dealing with contemporary mathematical topics (for mathematics students or for prospective or in-service mathematics teachers) or as enrichment for other courses. It can also be read with pleasure on its own by anyone interested in the intellectually intriguing aspects of mathematics
The teaching and learning of mathematics at university level an ICMI study by Derek Allan Holton(
) 22
editions published
between
1899
and
2010
in
English
and held by
492 WorldCat member
libraries
worldwide
This book No previous book has attempted to take such a wide view of the topic
Mathematical vistas : from a room with many windows by Peter Hilton(
Book
) 14
editions published
between
2000
and
2011
in
English
and held by
357 WorldCat member
libraries
worldwide
The goal of Mathematical Vistas is to stimulate the interest of bright people in mathematics. The book consists of nine related mathematical essays which will intrigue and inform the curious reader. In order to offer a broad spectrum of exciting developments in mathematics, topics are treated at different levels of depth and thoroughness. Some chapters can be understood completely with little background, others can be thought of as appetizers for further study. A number of BREAKS are included in each chapter. These are problems designed to test the reader's understanding of the material thus far in the chapter. This book is a sequel to the authors's popular book "Mathematical Reflections" and can be read independently
The Petersen graph by Derek Allan Holton(
Book
) 14
editions published
between
1990
and
2004
in
English
and held by
290 WorldCat member
libraries
worldwide
A first look at graph theory by John Clark(
Book
) 15
editions published
between
1991
and
2005
in
English
and held by
194 WorldCat member
libraries
worldwide
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This lesson unit is intended to help you assess how well students use algebra in context, and inparticular, how well students:
•
Explore relationships between variables in everyday situations.
•
Find unknown values from known values.
•
Find relationships between pairs of unknowns, and express these as tables and graphs.
•
Find general relationships between several variables, and express these in different ways byrearranging formulae.
COMMON CORE STATE STANDARDS
This lesson relates to the following
Standards for Mathematical Content
in the
Common Core StateStandards for Mathematics
:8.F Construct a function to model a linear relationship between two quantities.This lesson also relates to the following
Standards for Mathematical Practice
in the
Common CoreState Standards for Mathematics
:2. Reason abstractly and quantitatively.4. Model with mathematics.
INTRODUCTION
This activity links several aspects of algebra. A situation is presented, and relationships between thevariables are explored in depth. Letters are used to represent unknowns, generalized numbers, andvariables. The problem-solving context gives you a chance to assess how well students are able tocombine and apply different aspects of their algebra knowledge and skills.
•
Before the lesson, students work individually on the assessment task
The Guitar Cla
s
s
. You thenreview their work and create questions for students to answer at the end of the lesson, to helpthem to improve their solutions.
•
During the lesson, students translate between words, algebraic formulae, tables, and graphs in aninteractive whole-class discussion. The intention is that you focus students on making sense of the context using algebra, rather than just the routine use of techniques and skills. Students thenwork in pairs to graph the relationship between two of the variables. In a final whole-classdiscussion, students identify general formulae showing the relationships between all the variables.
•
At the end of the lesson, students review and improve their individual work.
MATERIALS REQUIRED
•
Each student will need two copies of the assessment task
The Guitar Class
, either the lesson tasksheet
Making and Selling Candles,
or the lesson task sheet
Rescue Helicopter
, depending on theoutcome of the assessment task, a mini-whiteboard, a pen, and an eraser.
•
Each pair of students will need a sheet of graph paper.
•
There are some projector resources to support whole-class discussion.
TIME NEEDED
15 minutes before the lesson and a one-hour lesson. All timings are approximate. Exact timings willdepend on the needs of the class.
Teacher guide Modeling Situations with Linear Equations T-2
BEFORE THE LESSON
Assessment task:
The Guitar Class
(15 minutes)
Have the students do this task in class or for homework, a day or more before the formativeassessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will beable to target your help more effectively in thefollow-up lesson.Give out
The Guitar Class.
Introduce the taskbriefly, and help the class to understand theproblem and its context.
Read through the questions, and try to answerthem as carefully as you can.What does 'profit' mean? Don't worry too much if you can't understand and do everything. I will teach a lesson with a task like this[tomorrow]. By the end of that lesson, your goal is to answer the questions with confidence.
It is important that, as far as possible, students are allowed to answer the questions without yourassistance.
Assessing students' responses
Collect students' responses to the task. Make some notes on what their work reveals about theircurrent levels of understanding. The purpose of this is to forewarn you of issues that will arise duringthe lesson itself, so that you may prepare carefully.We suggest that you do not score students' work. Research suggests this will be counterproductive, asit encourages students to compare their scores and distracts their attention from what they can do toimprove their mathematics.Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn fromcommon difficulties observed in trials of this lesson unit.We suggest that you write a list of your own questions, based on your students' work, using the ideasthat follow. You may choose to write questions on each student's work. If you do not have time to dothis, just select a few questions that will be of help to the majority of students. These can be writtenon the board at the end of the lesson.
Choosing the lesson task
After writing your list of questions, use your assessment of students' current understanding to decidewhich task to use during the lesson. We have found that many students learn from the
Making and Selling Candles
task.
However, if the majority of your students have answered most of the assessmenttask questions correctly, set the
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Find a Holden, MA Precal
...Instead of using numbers algebra introduces letters as symbols to represent generalized numbers that may vary(Variables). Algebra also defines the rules of mathematical expressions and equations. To me problem solving is like to solve a puzzle. You list the information you know and use variables for unknown information.
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9780321789433ugopolski's Precalculus: Functions and Graphs, Fourth Edition,gives you the essential strategies you need to make the transition to calculus. Throughout this book, you will find carefully placed learning aids and review tools to help them learn the math without getting distracted. The new edition includes over 900 additional exercises that are specifically designed to increase student understanding and retention of the concepts. Along the way, you'll see how the algebra connects to your future calculus course, with tools like Foreshadowing Calculusand Concepts of Calculus. Dugopolski's emphasis on problem solving and critical thinking helps you be successful in this course, as well as in future calculus courses.
Author Biography
Mark Dugopolski was born in Menominee, Michigan. After receiving a B.S. from Michigan State University, he taught high school for four years, and then went on to receive an M.S. in mathematics from Northern Illinois University. He also received a Ph.D. in the area of topology from the University of Illinois at Champaign-Urbana. Mark has been teaching at Southeastern Louisiana in Hammond, LA, ever since. Mark has been writing textbooks for about fifteen years. He is married and has two daughters, and enjoys playing tennis, jogging, and riding his bicycle in his spare time.
Table of Contents
P. Prerequisites
P.1 Real numbers and Their Properties
P.2 Integral Exponents and Scientific Notation
P.3 Rational Exponents and Radicals
P.4 Polynomials
P.5 Factoring Polynomials
P.6 Rational Expressions
P.7 Complex Numbers
Chapter P Highlights
Chapter P Review Exercises
Chapter P Test
1. Equations, Inequalities, and Modeling
1.1 Equations in One Variable
1.2 Constructing Models to Solve Problems
1.3 Equations and Graphs in Two Variables
1.4 Linear Equations in Two Variables
1.5 Scatter Diagrams and Curve Fitting
1.6 Complex Numbers
1.7 Quadratic Equations
1.8 Linear and Absolute Value Inequalities
Chapter 1 Highlights
Chapter 1 Review Exercises
Chapter 1 Test
Concepts of Calculus: Limits
2. Functions and Graphs
2.1 Functions
2.2 Graphs of Relations and Functions
2.3 Families of Functions, Transformations, and Symmetry
2.4 Operations with Functions
2.5 Inverse Functions
2.6 Constructing Functions with Variation
Chapter 2 Highlights
Chapter 2 Review Exercises
Chapter 2 Test
Tying it all Together
Concepts of Calculus: Instantaneous Rate of Change
3. Polynomial and Rational Functions
3.1 Quadratic Functions and Inequalities
3.2 Zeroes of Polynomial Functions
3.3 The Theory of Equations
3.4 Miscellaneous Equations
3.5 Graphs of Polynomial Functions
3.6 Rational Functions and Inequalities
Chapter 3 Highlights
Chapter 3 Review Exercises
Chapter 3 Test
Tying it all Together
Concepts of Calculus: Instantaneous Rate of Change of the Power Functions
4. Exponential and Logarithmic Functions
4.1 Exponential Functions and Their Applications
4.2 Logarithmic Functions and Their Applications
4.3 Rules of Logarithms
4.4 More Equations and Applications
Chapter 4 Highlights
Chapter 4 Review Exercises
Chapter 4 Test
Tying it all Together
Concepts of Calculus: The Instantaneous Rate of Change of f(x)= ex
5. The Trigonometric Functions
5.1 Angles and Their Measurements
5.2 The Sine and Cosine Functions
5.3 The Graphs of the Sine and Cosine Functions
5.4 The Other Trigonometric Functions and Their Graphs
5.5 The Inverse Trigonometric Functions
5.6 Right Triangle Trigonometry
Chapter 5 Highlights
Chapter 5 Review Exercises
Chapter 5 Test
Tying it all Together
Concepts of Calculus: Evaluating Transcendental Functions
6. Trigonometric Identities and Conditional Equations
6.1 Basic Identities
6.2 Verifying Identities
6.3 Sum and Difference Identities
6.4 Double-Angle and Half-Angle Identities
6.5 Product and Sum Identities
6.6 Conditional Trigonometric Equations
Chapter 6 Highlights
Chapter 6 Review Exercises
Chapter 6 Test
Tying it all Together
Concepts of Calculus: Area of a Circle and π
7. Applications of Trigonometry
7.1 The Law of Sines
7.2 The Law of Cosines
7.3 Vectors
7.4 Trigonometric Form of Complex Numbers
7.5 Powers and Roots of Complex and Numbers
7.6 Polar Equations
Chapter 7 Highlights
Chapter 7 Review Exercises
Chapter 7 Test
Tying it all Together
Concepts of Calculus: Limits and Asymptotes
8. Systems of Equations and Inequalities
8.1 Systems of Linear Equations in Two Variables
8.2 Systems of Linear Equations in Three Variables
8.3 Nonlinear Systems of Equations
8.4 Partial Fractions
8.5 Inequalities and Systems of Inequalities in Two Variables
8.6 The Linear Programming Model
Chapter 8 Highlights
Chapter 8 Review Exercises
Chapter 8 Test
Tying it all Together
Concepts of Calculus: Instantaneous Rate of Change and Partial Fractions
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Mathematics
Professors Goloubeva and Williams
Mathematics is an analytical tool used in all science and engineering courses. At the same time, by its very nature, mathematics is an abstract science. Mathematics at Webb is presented with the focus on applied mathematics, a branch of mathematics which is drawing on the physical world for its motivation, developing abstract concepts to refine the physical ideas, and finally applying those abstractions to mathematical modeling and better understanding of the phenomena of nature. Many Webb students go on to graduate work involving higher mathematics, and it is a strong objective of the mathematics program to prepare them well for this work.
Freshman Year
MATHEMATICS I - CALCULUS I
This is an introductory course whose main goals are to fill in the background of students who have already had an exposure to calculus in high school, to deepen their understanding of the material, and to develop their ability for abstract reasoning and mathematical modeling. The course starts with a discussion of vectors in the plane and in space, and basic vector operations, including dot and cross products. The course continues with a review of the real number system and inequalities, algebra of complex numbers, and the theory of elementary functions such as exponentials, logarithms, trigonometric functions, inverse trigonometric functions, hyperbolic functions and their inverses. The topics covered include limits, continuity, derivatives of functions of one variable, application of derivatives to curve sketching and to simple real life problems involving related rates, and optimization. The mean value theorem is covered. Linear and Taylor polynomial approximations are discussed, and applied to limits via L'Hopital's rule. The course includes a discussion of basic numerical methods such as method of bisections and Newton's method. The course concludes with a brief discussion of integration. To develop students' ability for abstract reasoning and to reach a deeper understanding of the material, the discussion often includes proofs.The pace is brisk.
The class meets four hours per week in the first semester.
MATHEMATICS II - CALCULUS II
The course starts with a discussion of integration, integration techniques and applications of integrals. The topics of discussion include the Riemann Integral, the review of the basic techniques of integration such as substitution, integration by parts, partial fraction decomposition and trigonometric techniques of integration. The course covers applications of definite integrals to simple problems involving area between curves, arc length, volume, projectile motion, work and center of mass. The course continues with a discussion of the theory of parametric equations, plane curves, and polar coordinates. The concepts of calculus are extended to curves described by parametric equations and polar coordinates. In this course the geometry of three-space is covered more extensively than in Mathematics I. Cylindrical and spherical coordinates are introduced. Emphasis is placed on visualization and graphical representation of surfaces in space.
This course contains most of the calculus of functions of several variables and includes concepts of limits and continuity of functions of several variables, partial derivatives, tangent planes and linear approximations, gradients, differentials and directional derivatives. In this course we introduce the mathematical basis for finding the maximum or minimum of functions of several variables. Optimization problems for functions of several variables are introduced. The discussion includes constrained optimization and the method of Lagrange multipliers. The course also includes a brief introduction to linear algebra, which covers the rudiments of matrix algebra. Determinants are also introduced here. The course concludes with a unified discussion of real infinite series.
The class meets four hours per week in the second semester.
Sophomore Year
MATHEMATICS III - DIFFERENTIAL EQUATIONS
This class is a basic course in differential equations. It starts with classification of differential equations. It continues with the discussion of ordinary differential equations (ODEs), starting with first order ordinary differential equations. The concepts of direction fields, boundary, and initial value problems are introduced. Several methods of solution of first-order ODEs are considered. It is emphasized that each method is applicable to a certain subclass of first-order equations. The topics covered include methods of solutions of linear equations, separable equations, homogeneous and exact equations, Bernoulli and autonomous equations. The main methods under discussion are the integration of factors, variation of parameters, and separation of variables. The idea of approximating a solution by numerical computation is introduced in the discussion of Euler's method.
The course continues with a discussion of the general theory and methods of solution of second and nth order ordinary differential equations. Conditions for the existence and uniqueness of the solution are analyzed. Substantial attention is given to methods of solution of second-order differential equations with constant coefficients. Methods under discussion are the reduction of order, undetermined coefficients, the variation of parameters, series solutions, and the Laplace transform.
The course concludes with a discussion of partial differential equations, Fourier series, and separation of variables as a method for solving partial differential equations. Throughout the course, applications of differential equations to simple physical problems are thoroughly discussed.The class meets three hours per week in the first semester.
MATHEMATICS IV - ADVANCED ENGINEERING MATHEMATICS
There are essentially three separate components of the course. The first component involves a discussion of multiple integrals and vector calculus. This material can best be described as the mathematics needed to study fluids. The course covers the theory of vector-valued functions. Multiple integrals are covered extensively. Emphasis is placed on transformation of space/coordinates and the role of the Jacobian. The concepts of vector and scalar fields, curl, and divergence are introduced from a very physical point of view, as are line and surface integrals. The three major theorems of vector calculus - Green's theorem, Stoke's theorem, and the Divergence (Gauss') theorem - are covered. A strong emphasis is placed on physical interpretation. This material is highly visual and makes extensive use of Maple to illustrate the concepts.
The second component of this class involves complex variables. This component covers the basic arithmetic and geometry of the complex number system. Then the calculus of functions of complex numbers is studied, including the Cauchy-Riemann equations and the implications for harmonic functions. Complex exponential, trigonometric, and logarithmic functions are defined and studied. There is a brief treatment of conformal mapping. In addition, standard integral procedures are discussed.
The third component of this course covers the remaining essential parts of linear algebra and differential equations. The class works more extensively with matrices, matrix functions, and the calculus of matrix functions. Then it discusses methods of solution of systems of first-order linear equations, eigenvalues and eigenvectors, and methods of solution of systems of differential equations. The course meets four hours per week in the second semester.
Junior Year
PROBABILITY AND STATISTICS
This course begins with an introduction to probability theory, including set theoretic and combinatorial concepts. This is followed by treatments of discrete random variables and distributions and continuous random variables. Generating functions are discussed at some length. Particular emphasis is placed on the Rayleigh and Weibull distributions, which are applied subsequently in the Ship Dynamics course as models of wave spectra and are also encountered as models of the manufacturing process. The latter third of this course addresses the application of statistical methods to engineering experimentation, beginning with an introduction to estimation and hypothesis testing and culminating with an overview of experiment design. The course meets four hours per week in the first semester.
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7th Grade Math Syllabus
Course Format: Lecture, class discussion, cooperative learning, tests, quizzes,
projects, and daily grades.
Location: Riverton Middle School, Room 217. To find my room come in the double
doors leading to the seventh grade hall. Take the hall on your left. Take the first left
and keep going until you are able to see my room. My room is located besides Mrs.
Douglass and Mrs. Hernandez.
Supplies: 3-Ring binder, paper, pencils, calculator (optional)
Instructor: Samuel Wallace Email: swallace@madison.k12.al.us
You can reach me anytime by my email. It will go straight to my Blackberry. If you
have any questions about assignments or problems with homework you can reach
me there.
Attendance Policy: Per state law attendance in mandatory.
Grading Policy:
Daily Quiz 10%
Homework 15% Daily Grades
Quizzes 15%
Tests 60%
Total 100%
Description of Assignments:
Daily Quiz – This will be given every day. It is simply one problem each day based on
a previous day's lecture or homework. Each daily quiz will be worth 5 points for a
total of 25 at the end of each week. These are graded on the following symbols:
This means the answer to the daily quiz is incorrect.
A This means the student failed to execute the next
appropriate act, such as walking back to his/her seat quietly.
H This means that the student did not complete the previous
night's homework.
T This means that the student was talking before all daily
quizzes were handed in.
L This means the student was late to class.
For example: If a student turns in his/her daily quiz for the day and he/she does all
the above correctly, then he/she will receive all 5 points for that day. The weekly
grade is calculated by finding the percentage of points earned to total points
possible.
Homework – Homework is an essential part of learning mathematics. The best way
to learn math is through the end of a pencil. So, most nights will include homework
in my class. Each homework assignment is worth 5 points. Homework will be
checked each day while the daily quiz is being given. Homework will be graded on
the following scale:
5 – All homework was completed
3 – At least half of the homework was completed
0 – Less than half of the homework was completed
Quizzes – Formal evaluations on sections or portions of units of instruction.
Tests – Formal evaluations on entire units of instruction.
Discipline Policy:
Class Rules:
1. Students will be in his/her seat when the bell rings.
2. Students will bring all supplies to class.
3. Students will act respectfully at all times toward the teacher, other students,
and themselves.
Consequences: If the student violates one of the rules, they will receive an offense.
The following is a list of consequences based on the number of offenses earned:
1st offense – Warning
2nd offense – Writing assignment written 4 times.
3rd offense – Writing assignment written 8 times.
4th offense – Parent/teacher conference requested.
5th offense – Referred to the office.
Offenses are reset at the end of each week. If the student has received 0 offenses
for the week he/she will be allowed to draw from the reward jar. The reward jar
contains items such as: 3 points on a test, 5 points on a quiz, free pencil, etc.
Makeup Work: Students who are absent should check the makeup folder for his/her
particular period. The folder contains lecture notes from the previous day and
homework assignments that the student might have missed. It is the student's
responsibility to find his/her makeup work and turn it in to the teacher.
Other classroom policies:
The teacher dismisses class not the bell.
Fire escapes and inclement weather routes are posted and
will be practiced.
No gum is allowed in the classroom.
The student is required to sit in his/her assigned seat.
All papers will be turned into the appropriate period tray.
Students must remain in his/her seat unless called upon.
Bathroom trips are not allowed unless it is an emergency.
A student can choose to take an offense to go to the
restroom.
I have read the syllabus above and fully understand what is expected for
__________________________ to succeed in Coach Wallace's class.
Student's Name
Parent Signature
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Thank you for visiting us. We are currently updating our shopping cart and regret to advise that it will be unavailable until September 1, 2014. We apologise for any inconvenience and look forward to serving you again.
Author Information
Mary Jane Sterling is the author of four other For Dummies titles: Algebra For Dummies, Algebra II For Dummies, Trigonometry For Dummies, and Math Word Problems For Dummies. She has honed her math-explaining skills during her years of teaching mathematics at all levels: junior high school, high school, and college. She has been teaching at Bradley University, in Peoria, Illinois, for almost 30 of those years.
When not teaching or writing, Mary Jane keeps busy by working with her Kiwanis Club, advising Bradley University's Circle K Club, and working with members of the Heart of Illinois Aktion Club (for adults with disabilities). All the volunteer projects taken on for these clubs help keep her busy and involved in the community.
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Realm of measure.
This book combines the history of mathematics with an explanation of mathematical procedures and principles.This book combines the history of mathematics with an explanation of mathematical procedures and principles
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Choose your search criteria in the right column.
lecture
This course explores the interconnectedness of math and science concepts and principles, engaging in hands-on units to prepare teachers to help students explore and make sense of the world of science through the application of mathematical principles, and vice versa.
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Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for successN/a Boston, MA 2011 Soft Cover 4th Edition Good 4to. No writing, highlighting, or creases. Cover has edgewear, an upcurl, & a yellow stain on the fore-edge & bottom corner. First few pages also have...show more a faint stain on the bottom corner. [#54328] ...show less
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's popular Pre-GED Satellite Series targets students with reading levels of 5-8 and delivers instruction in the GED content areas to prepare learners to move up into GED preparation. The pre- and post-tests in each satellite book help students decide which skills need the most work and allow them to assess mastery of each subject area. Complete answer keys supply answers and explanations for all the exercises and activities within the text and lots of visual aids provide detailed practice and instruction. Pre-GED Mathematics will give your students the basic skills needed to succeed in their study toward a GED. The book begins with adding whole numbers and other simple operations. Problem solving is a skill students will spend extra time on through activities and exercises. Pre-GED Mathematics provides many real-life problems with easy application to everyday usage. Alternative format responses are also included in this text as well as activities for the Casio fx-260, the only calculator that can be used on the GED exam.
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Conventional calculus is too hard and too complex. Students are forced to learn too many theorems and proofs. In Free Calculus, the author suggests a direct approach to the two fundamental concepts of calculus — differentiation and integration — using two inequalities. Regular calculus is condensed into a single concise chapter. This... more...
This book is a detailed study of Gottfried Wilhelm Leibniz's creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known "calculi" Leibniz created such as the Analysis Situs,... more...
In its largest aspect, the calculus functions as a celestial measuring tape, able to order the infinite expanse of the universe. Time and space are given names, points, and limits; seemingly intractable problems of motion, growth, and form are reduced to answerable questions. Calculus was humanity's first attempt to represent the world and perhaps... more...... more...
Your INTEGRAL tool for mastering ADVANCED CALCULUS
Interested in going further in calculus but don't where to begin? No problem! With Advanced Calculus Demystified , there's no limit to how much you will learn.
Beginning with an overview of functions of multiple variables and their graphs, this book covers the fundamentals, without spending... more...
Packed with practical examples, graphs, and Q&As, this complete self-teaching guide from the best-selling author of Algebra Demystified covers all the essential topics, including: absolute value, nonlinear inequalities, functions and their graphs, inverses, proportion and ratio, and much more. more...
LEARNING CALCULUS JUST GOT A LOT EASIER! Here?s an innovative shortcut to gaining a more intuitive understanding of both differential and integral calculus. In Calculus Demystified an experienced teacher and author of more than 30 books puts all the math background you need inside and uses practical examples, real data, and a totally different... more...
Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner,... more...
Boost Your grades with this illustrated quick-study guide. You will use it from high school all the way to graduate school and beyond. Includes both Calculus I and II. Clear and concise explanations. Difficult concepts are explained in simple terms. Illustrated with graphs and diagrams. Search for the words or phrases. Access the guide anytime, anywhere... more...
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Throughout the years as an engineer, I have needed to research topics on engineering, physics, chemistry, mechanics, mathematics, etc. The Internet has made the job infinitely simpler, with the caveat that you have to be careful of your sources. Anyo...
Find articles and problems on the site by choosing curriculum topics.
Each problem page displayed gives a list
of similar problems and a list of topics to help you find more.
Alternatively see our previous .
Olympic National Park in Washington offers a bit of everything for nature lovers, hikers and adventurers like Pacific Ocean beaches, rain forest valleys, glacier-capped peaks and a dazzling diversity of plants and animals. 95% of this park is still w...
, by is a spectacular introduction to the world of electronics. This book is not new - the truth is that it has changed little since it's first release in 1983. Despite this, twenty-five years later, there is really nothing else like it. This boo...
This post originally appered in Business Insider. Mathematics is all around us, and it has shaped our understanding of the world in countless ways. In 2013, mathematician and science author Ian Stewart published a book on "17 equations that changed t...
The TI-83plus graphing calculator is used extensively in our AP Chemistry program. It is used in conjunction with the lab component, where we do many labs with the CBL and Vernier probes. In addition, many calculations can be simplified (once the the...
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Properties of exponents:
Multiplication of exponents rule is defined as:
Division of exponents rule is defined as:
The power rule of exponents is defined as:
Interval notation:
Interval notation is another form of representing inequalities without having to use too much mathematical notationIntermediate Algebra prerequisites are covered in this tutorial. The use of real numbers and what operations are performed in example problems are introduced in this tutorial.
A representation of inequalities using interval notation and the use of absolute values are presented in this tutorial. Important mathematics principles such as scientific notation and formulas are shown here to emphasize the importance of them in mathematics application problems. Scientific notation is commonly used in course such as chemistry and biochemistry to represent values.
Series Features:
• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
• Definition slides introduce terms as they are needed.
• Visual representation of concepts
• Animated examples—worked out step by step
• A concise summary is given at the conclusion of the tutorial.
"Title" Topic List:
Inequalities and their interval notation representation Absolute Values and Examples The operations of real numbers Algebraic Expressions and Examples The properties of exponents Scientific Notation Formulas and Examples
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Updated to reflect the newest version of MATLAB, this edition features a new layout for enhanced readability. The book covers both linear and nonlinear difference equations, elementary functions, numerical differentiation, integration and ordinary differential equations solving techniques, optimization methods, complex numbers, vectors, matrix algebra and special matrices, geometric and Lorentz transformations, and probability theory.
New to the Second Edition:
Updated MATLAB syntax that conforms to MATLAB 7.1
Expanded introductory chapter that reduces the need to refer to MATLAB online help or user manuals
Special advanced sections for students looking for more challenging material
Appendix of symbolic capabilities of MATLAB
Smoothing the transition from elementary math, physics, and computer science sequences to more advanced engineering concepts, this book helps students master fundamental quantitative tools that allow them to progress to more complex electrical and computer engineering applications.
DIFFERENCE EQUATIONS
Simple Linear Forms
Amortization
An Iterative Geometric Construct: The Koch Curve
Solution of Linear Constant Coefficients Difference Equations
Convolution-Summation of a First-Order System with Constant Coefficients
General First-Order Linear Difference Equations
Nonlinear Difference Equations
Fractals and Computer Art
Generation of Special Functions from Their Recursion Relations
ELEMENTARY FUNCTIONS AND SOME OF THEIR USES
Function Files
Examples with Affine Functions
Examples with Quadratic Functions
Examples with Polynomial Functions
Examples with the Trigonometric Functions
Examples with the Logarithmic Function
Examples with the Exponential Function
Examples with the Hyperbolic Functions and Their Inverses
Commonly Used Signal Processing Functions
Animation of a Moving Rectangular Pulse
Use of the Function Handle
MATLAB Commands Review
ROOT SOLVING AND OPTIMIZATION METHODS
Finding the Real Roots of a Function of One Variable
Roots of a Polynomial
Optimization Methods for Functions of One Variable
The Zeros and the Minima of Functions in Two Variables
Finding the Minima of Functions with Constraints Present
MATLAB Commands Review
COMPLEX NUMBERS
Introduction
The Basics
Complex Conjugation and Division
Polar Form of Complex Numbers
Analytical Solutions of Constant Coefficients ODE
Phasors
Interference and Diffraction of Electromagnetic Waves
Solving ac Circuits with Phasors: The Impedance Method
Transfer Function for a Difference Equation with Constant Coefficients
MATLAB Commands Review
MATRICES
Setting up Matrices
Adding Matrices
Multiplying a Matrix by a Scalar
Multiplying Matrices
Inverse of a Matrix
Solving a System of Linear Equations
Application of Matrix Methods
Eigenvalues and Eigenvectors
The Cayley-Hamilton and Other Analytical Techniques
Special Classes of Matrices
Transfer Matrices
Covariance Matrices
MATLAB Commands
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From June through August, we are continuing to post problems every two weeks; and members may still submit solutions. We encourage teachers with class, school, or district subscriptions to take this opportunity to have their students do math all summer. Design and assign summer projects using the Current Problems or from a selection from the Problems of the Week Library.
We will have only a few volunteers doing mentoring; and our responses will otherwise be scaled back as we get ready for a new series of problems at the end of August. Check the calendar link on each PoW page for a full summer schedule.
Sadie Bowman and Marc Gutman keep a blog of their performances in "Calculus: The Musical!" This comic "review" blend of sketch comedy, musical theatre, and lecture about the concepts and history of calculus emerged as a teaching tool from the classroom of Gutman, who "... found that setting formulas and rules to music helped his students learn and retain tricky information.
"'Maxima' and 'minima' is an abstract concept to a lot of us, but when sung as a rousing 'Can-Can' chorus, it's fun and easy to remember!" Using musical parodies that span genres from light opera to hip hop, Bowman and Gutman introduce and illuminate such concepts as limits, integration and differentiation, dramatizing some high points of calculus' history. Musical tributes reference The Beatles, Gilbert & Sullivan, Madonna, Petula Clark and Eminem.
Through the website book their act, check out their tour calendar, read critics' reviews, preview lyrics, purchase the audio CD, watch video, and more
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Students describe graphically, algebraically and verbally real-world phenomena as functions; identify the independent and the dependent variable. They translate among graphic, algebraic, and verbal representations of relations and graph relations and functions and find the zeros of functions. Finally, students write and interpret an equation of a curve (linear) which models a set of data.
Students explore the concept of linear regression. In this linear regression line lesson, students make a scatter plot using world population data over the course of 30 years. Students find the line of regression to fit the data. Students find the derivative of the data.
High schoolers investigate linear data. In this linear data lesson, students collect data about their weight. High schoolers weigh themselves on a board at various distances from the scale. Students find a linear model to fit their data. High schoolers use their model to predict the weight on the scale of a person standing a given distance away from the scale.
Learners model and record subtraction using the "take away" method with numbers 0 to 10 as addends. They also find missing addends. Students will interpret the y-intercept in the context of "real-world" data.
Students explore the concept of scatterplots. In this scatterplots lesson, students enter temperature and humidity data into their calculator. Students plot the points using a scatterplot. Students discuss the relationship between temperature and humidity.
Playing with matches (unlit, of course) becomes an engaging learning experience in this fun instructional unit. Teach pupils how to apply properties of exponential functions to solve problems. They differentiate between quadratic and exponential functions in a series of hands-on lessons that include worksheets, assessments, and answer keys.
Ninth graders explore regression equations. In this Algebra I lesson, 9th graders create lists of data points and determine the regression equation of the line that best fits the data points. In this second part to the lesson, students investigate the mathematics behind how the linear regression is determined.
Eleventh graders explore quadratic equations. In this Algebra II lesson, 11th graders examine the motion of a bouncing ball. Students collect, graph, and model data. Both the standard and vertex form of the quadratic equation are considered.
Students find a quadratic model to fit Olympic swim times. In this finding a quadratic model to fit Olympic swim times lesson, students enter data into lists. Students create a scatter plot of the Olympic swim data. Students find a quadratic model to fit the data by performing a quadratic regression.
Students explore several examples of cardiovascular diseases. In this anatomy lesson, students explain why physical fitness is very important. They count their pulse rate and record them on a data table.
Young scholars explore the concept of density. In this density lesson, students measure the density of pennies using water, pennies made before 1982 and after 1982. Young scholars collect data on their graphing calculator using a force probe. Students graph the data and determine the line of best fit.
Students collect data, analyze their data and draw conclusion. In this statistics instructional activity, students identify different patterns through graphing. They make predictions using these patterns and the line of best fit for the future. They approximate the line of best fit using two points.
Study various types of mathematical models in this math activity. Learners calculate the slope to determine the risk in a situation described. They respond to a number of questions and analyze their statistical data. Then, they determine and graph a curve that best fits each situation given. Additionally, they log on to two different sites and use the tool and data provided to find the elevation of a given point selected.
Students study practical data analysis within the constraints of the scientific method. In this data instructional activity students collect and enter data into a computer spreadsheet then create graphs.
Students investigate the centroid of a data set and its significance for the line fitted to the data. They investigate the relationship between a set of data points and a curve used to fit the data points.
Students study how to monitor their personal status of their body composition. They study how to monitor and adjust activity levels to meet personal fitness needs and demonstrate objectives 1 and 2 by using the software provided by Furtex.
Learners explore the properties of enzymes. In this chemical reaction lesson, students explore enzymes through a Web-quest and investigative study. Learners will collect and summarize data and create a class presentation. This lesson is hands on and includes multiple web links.
Students determine the true cost of owning a car. In this determining the true cost of owning a car lesson, students examine the true cost of owning a car. Students calculate the cost of buying a car, insurance, gas, etc. Students make a scatter plot of a given cars' mpg vs. speed. Students find a quadratic model to fit the data.
Students investigate the causes of heart disease and survey the staff to gather data on their health and family history. In this health instructional activity students create a PowerPoint presentation and brochure on the communities health.
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second edition of this hands-on workbook presents exercises, problems, and quizzes with solutions and answers as it takes college-bound students through all math and science topics covered on the ACT. Separate math chapters cover:Pre-algebra, elementary algebra, and intermediate algebraPlane geometry, coordinate geometry, and trigonometryThe science sections emphasize the scientific method and focus on how to read scientific passages. Science topics covered include:Data representation passagesResearch summaries passagesConflicting viewpoints passages Additional features include a glossary of science terms and test-taking strategies for success. The workbook concludes with a full-length math and science practice test.
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Linear Systems
Linear systems have all the necessary elements (modeling, identification, analysis and control), from an educational point of view, to help us understand the discipline of automation and apply it efficiently. This book is progressive and organized in such a way that different levels of readership are possible. It is addressed both to beginners and those with a good understanding of automation wishing to enhance their knowledge on the subject. The theory is rigorously developed and illustrated by numerous examples which can be reproduced with the help of appropriate computation software. 60 exercises and their solutions are included to enable the readers to test and enhance their knowledge.
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GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On...
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GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.The GeoGebraWiki is a free pool of educational materials for GeoGebra. Everyone can contribute and upload materials there: International GeoGebraWiki - pool of educational materials for GeoGebra and the German GeoGebraWiki The Dynamic Worksheets GeoGebra can also be used to create dynamic worksheets:Pythagorasvisualisation of Pythagoras' theoremLadder against the Wallapplication of Pythagoras' theorem Circle and its Equationconnection between a circle's center, radius and equation Slope and Derivative of a Function (3 sheets)relation between slope, derivative and local extrema of a functionDerivative of a Polynomial interactive exercise to practice finding the derivative of a cubic polynomialUpper- and Lower Sums of a Functionvisualisation of the backgrounds of Riemann's Integral
The Math Solutions Newsletter contains over 20 innovative hands-on activities for teaching a range of mathematics concepts in...
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The Math Solutions Newsletter contains over 20 innovative hands-on activities for teaching a range of mathematics concepts in elementary and middle grades math methods courses. Math Solutions is lead by Marilyn Burns, one of today's most highly respected mathematics educators. For more than 30 years, she has been teaching and writing for children, leading inservice workshops, and creating teacher resource materials.
Building and exploring mathematical models is a fundamental task in science. MODELLUS offers students and teachers a...
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Building and exploring mathematical models is a fundamental task in science. MODELLUS offers students and teachers a "minds-on״, multilevel learning experience in which they create, simulate, and analyze models interactively on the computer, either from experimental data and images or pure from theoretical thinking.Modellus is software for interactive modelling with mathematics. Teachers and students can use Modellus to build mathematical models and explore them as animations, graphs, and tables.Instead of just looking at algebraic, differential, and iterative equations, Modellus users can experiment visually and interactively with models and animations to better understand the underlying mathematics and the multiple representations of a model.Modellus can also be used as a tool to analyse and make sense of experimental data, providing tools to maake models from images (BMP or GIF) and videos (AVI)Modelles have won educational prices aswell in the USA as in Europe. In Great Britain is is used bij hundreds of teachers and thousands of students in the Advancing Physics Project.Best of all: It is free for educational use!I found this one of the most valuable pieces of educational software I have seen last few years.At the website you find the software and manual (downloadable) and lot of other information (examples, papers, tutoriala, etc.)
Excellent computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels....
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Excellent computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels. This is an NSF supported project that began in 1999 to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-12 emphasis). Free!
Students will attempt to solve a mystery by infiltrating a secret society, answering initiation questions regarding history,...
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Students will attempt to solve a mystery by infiltrating a secret society, answering initiation questions regarding history, science, mathematics, world languages, and the arts. They will contact and be contacted by various fictitious characters via email, telephone, and instant messenger who will provide clues that enable them to continue their quest to discover the truth about a mysterious artifact known only as the Hexagon. Up to six extra credit points will be awarded along the way, which can be applied to your course if you so choose. The Hexagon Challenge encourages students to: 1) Use critical thinking skills. 2) Locate important information using Internet resources. 3) Utilize creative problem solving.
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College Algebra
David Cohen's COLLEGE ALGEBRA, Fifth Edition, focuses on teaching mathematics, using a graphical perspective throughout to provide a visual ...Show synopsisDavid Cohen's COLLEGE ALGEBRA, Fifth Edition, focuses on teaching mathematics, using a graphical perspective throughout to provide a visual understanding of college algebra. The author is known for his clear writing style and the numerous quality exercises and applications he includes in his respected texts. In this new edition, graphs, visualization of data, and functions are now introduced much earlier and receive greater emphasis. Many sections now contain more examples and exercises involving applications and real-life data. While this edition takes the existence of the graphing calculator for granted, the material is arranged so that one can teach the course with as much or as little graphing utility work as he/she wishesPoor. Shipping fast. No guarantee for ancillary materials(Such...Poor. Shipping fast. No guarantee for ancillary materials(Such as CDs, Online access code). Ships today or the
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This book has been written for a one-semester combined linear algebra and differential equations course, yet it contains enough material for a two-term sequence in linear algebra and differential equations. By introducing matrices, determinants, and vector spaces early in the course, the authors are able to fully develop the connections between linear algebra and differential equations. The book is flexible enough to be easily adapted to fit most syllabi, including courses that cover differential equations first. Technology is fully integrated where appropriate, and the text offers fresh and relevant applications to motivate student
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Chapter 3: Graphs, Linear Equations, and Functions 3.1 The Rectangular Coordinate System 3.2 The Slope of a Line 3.3 Linear Equations in Two Variables Summary Exercises on Slopes and Equations of Lines 3.4 Linear Inequalities in Two Variables 3.5 Introduction to Functions45 +$3.99 s/h
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AllAmericanTextbooks Ypsilanti, MI
0321442547 Cover worn but text is still usable. ISBN|0321442547 Algebra for College Students (6th Edition) (C.)2008 (RMV) WQGood
newrecycleabook centerville, OH
0321442547 used book - free tracking number with every order. book may have some writing or highlighting, or used book stickers on front or back
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Summary: Susanna Epp's DISCRETE MATHEMATICS, THIRD EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circui...show morets and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses. ...show less
2004
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p>This Saxon Math Homeschool 7/6 Tests and Worksheets book is part of the Saxon Math 7/6 curriculum for 6th grade students, and provides supplemental facts practice tests for each lesson, as well as 23 cumulative tests that cover every 5-10 lessons. The included activity sheets are designed to be used with the activities given in the (sold- separately) student worktext. A testing schedule and five optional, reproducible, recording forms are also provided; three forms allow students to record their work on the daily lessons, mixed practice exercises, and tests, while the remaining two forms help teachers track and analyze student performance. Solutions to all problems are in the (sold- separately) Solutions Manual. 241 newsprint-like, perforated pages, softcover. Three-hole-punched ...
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Author(s): Spencer A. Rathus |Publisher: Wadsworth Publishing | The smartest way to save money on your textbooks! Compare the prior version(pictured left) with ISBN: 049551041950'S NEW ORIGINAL CHAMPION ENGINE TUNE UP WORKSHEETS This is a Lot of 4 Original Worksheets for: Engine Tune-Ups Assorted Vehicles Champion Form No. A-757 Wow Rare!!! ABOUT US Jones Vintage Parts has been a family-owned business for over 30 years. We specialize in rare and vintage NOS and used parts for all makes and models from the teens to the 70s.. We have three (3) warehouses across the United States full of vintage cars and parts! New listings posted daily! SHIPPING INFORMATION Purchasedzypowekowygu store Algebra 2 Tutor Companion Worksheet CD This is the companion product designed to accompany the Algebra 2 Tutor video DVD. It is recommended that this product be used along side the Algebra 2 Tutor DVD to enhance mastery of Algebra 2. LOWEST PRICE ON THE INTERNET INCLUDING AMAZON AND THE MANUFACTURER'S WEBSITE HOMESCHOOL SEAL OF APPROVAL -- FAMILY REVIEW GOLD AWARD List Price is $34.99 MathTutorDVDs are the #1 Best Selling tutorial DVDs in their respective subjects! LOWEST
Free Delivery Worldwide : Alternatives to Worksheets : Paperback : Creative Teaching Press : 9781574714296 : 1574714295 : 01 Jul 2001 : Meaningful child-centered activities, simple directions, and a teacher-friendly format. 40 activities with hundreds of variations for quality seatwork. Alternatives to Worksheets is just what every teacher needs a resource book with hundreds of meaningful and motivating activities that students can work on independently. Now you and your students will never be at a loss for exciting and challenging projects.A welcome change of pace from worksheets, these activities will raise student involvement to new levels and generate genuine enthusiasm for learning. You will love the imaginative alternatives described in this book and so will your students
Author(s): Tobey, John Jr; Slater, Jeffrey; Blair, Jamie; Crawford, Jennifer |Publisher: Pearson | The smartest way to save money on your textbooks! Compare the prior version(pictured left) with ISBN: 0321748522 Early Phonics 4 Worksheets (c,s,a,t,m,n) by Ian Mitch and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
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eBayJesus The Way Bible Study With Leader And Student Worksheets Dvd
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Jesus The New Way Bible Study with Leader and Student Worksheets NEW DVD Click to Enlarge Jesus The New Way Brand NEW DVD Wright's acclaimed scholarship is delivered in a winsome and understandable way, showing how Jesus is the fulfillment of Israel's ancient hopes and humanity's deepest dreams. You will see Jesus as you have never seen him before in the context of his Jewish and Roman world. You will find explosive new meaning in his familiar words and deeds as Wright unfolds his incomparable
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Practice Makes Perfect PrecalculusDon't be perplexed by precalculus. Master this math with practice, practice, practice! Practice Makes Perfect: Precalculusis a comprehensive guide and workbook that covers all the basics of precalculus solution for getting a handle on math right away, Practice Makes Perfect: Precalculusis your ultimate resource for building a solid understanding of precalculus fundamentals.
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The Fundamentals of Thinking Well
Logic is foundational for helping students excel in every subject they study, from math and science to rhetoric and humanities. Completely revised and expanded, the Introductory Logic DVD course will teach students eighth grade and up the fundamentals of thinking well. This course isn't just a supplement for a logic class -- it brings a skilled logician with more than twenty years of teaching experience right into your home. James Nance walks you through every lesson in his bestselling Introductory Logic textbook: definitions, logical statements, fallacies, syllogisms, and much more, not to mention practice tests and other helps for those learning at home. Enjoy!
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The study of mathematics helps students to develop thinking skills, order their thoughts, develop logical arguments, and make valid inferences. The goal of all mathematics courses is to enable students to discern mathematical relationships, reason logically, and use mathematical techniques effectively to solve real-world problems.
Mission Statement
To ensure each student is placed where he/she has the tools necessary for success and to challenge all students to achieve their potential. Courses are standards-based and students master the standards prior to progressing.
Course List, Textbooks and text coverage
The Mathematics Department's Course List provides a listing and description of the classes at Lowell offered by the Mathematics Department along with prerequisites for prospective students. Here (updated August 2013) is a list of textbooks currently used with the sections covered by course semester.
The Mathematics Program
All entering ninth-grade students are tested for math placement. Placement is determined based on a combination of the mathematics placement test and/or the Algebra I California Standards Test that should be given in middle school. If students are unhappy with their initial placement, they may attempt to audit the higher course along with the course they were placed into and move into the more advanced course sequence should be successful in the audited course. Entering the honors path after Accelerated Math 2H is difficult because honors math courses move at a much more rapid pace than regular math courses.
Incoming 9th Grade Student Testing and Placement
When Lowell received supplemental test scores for students coming from SFUSD middle schools, we were also informed of a new Administrative Regulation which states that incoming 9th graders who have passed a middle school Algebra course and who score 60% or higher on the MDTP are to be placed in geometry or higher. While the previous regulation provided a range of values (60-70% MDTP scores) for Geometry admission, the new regulation does not.
There are several MDTP tests that measure Algebra skills, our placement test being one and the test given by the SFUSD middle schools being another. Accordingly, to be fair to all, any student scoring 27 – 30 (31 or above are already in Geometry or Accelerated Math) will be placed into Geometry. This does assume you have passed a middle school algebra class, which is subject to verification. I am sorry this was not the cutoff published in my May letter.
Still, Algebra is the foundation of future mathematics and if you/your student is weak (particularly in the low 60% range) you may want to consider repeating Algebra. A written request from a parent requesting Algebra will be honored over the district's Administrative Regulation.
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