Mathematica by Example This page intentionally left blank Mathematica by Example Fourth Edition Martha L Abell and James P Braselton Department of Mathematical Sciences Georgia Southern University Statesboro, Georgia AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper Copyright ∞ © 2009 by Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.com You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data APPLICATION SUBMITTED British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374318-3 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 09 10 11 12 Contents Preface ix CHAPTER Getting Started 1 1.1 Introduction to Mathematica A Note Regarding Different Versions of Mathematica 1.1.1 Getting Started with Mathematica Preview Five Basic Rules of Mathematica Syntax 1.2 Loading Packages 1.2.1 Packages Included with Older Versions of Mathematica 1.2.2 Loading New Packages 1.3 Getting Help from Mathematica Mathematica Help 1.4 Exercises 13 13 13 14 15 17 24 28 2.1 Numerical Calculations and Built-in Functions 2.1.1 Numerical Calculations 2.1.2 Built-in Constants 2.1.3 Built-in Functions A Word of Caution 2.2 Expressions and Functions: Elementary Algebra 2.2.1 Basic Algebraic Operations on Expressions 2.2.2 Naming and Evaluating Expressions 2.2.3 Defining and Evaluating Functions 2.3 Graphing Functions, Expressions, and Equations 2.3.1 Functions of a Single Variable 2.3.2 Parametric and Polar Plots in Two Dimensions 2.3.3 Three-Dimensional and Contour Plots: Graphing Equations 2.3.4 Parametric Curves and Surfaces in Space 2.3.5 Miscellaneous Comments 2.4 Solving Equations 2.4.1 Exact Solutions of Equations 2.4.2 Approximate Solutions of Equations 2.5 Exercises 31 31 31 34 35 38 39 39 44 47 52 52 65 71 82 94 100 100 110 115 CHAPTER Basic Operations on Numbers, Expressions, and Functions v vi Contents CHAPTER Calculus 3.1 Limits and Continuity 3.1.1 Using Graphs and Tables to Predict Limits 3.1.2 Computing Limits 3.1.3 One-Sided Limits 3.1.4 Continuity 3.2 Differential Calculus 3.2.1 Definition of the Derivative 3.2.2 Calculating Derivatives 3.2.3 Implicit Differentiation 3.2.4 Tangent Lines 3.2.5 The First Derivative Test and Second Derivative Test 3.2.6 Applied Max/Min Problems 3.2.7 Antidifferentiation 3.3 Integral Calculus 3.3.1 Area 3.3.2 The Definite Integral 3.3.3 Approximating Definite Integrals 3.3.4 Area 3.3.5 Arc Length 3.3.6 Solids of Revolution 3.4 Series 3.4.1 Introduction to Sequences and Series 3.4.2 Convergence Tests 3.4.3 Alternating Series 3.4.4 Power Series 3.4.5 Taylor and Maclaurin Series 3.4.6 Taylor’s Theorem 3.4.7 Other Series 3.5 Multivariable Calculus 3.5.1 Limits of Functions of Two Variables 3.5.2 Partial and Directional Derivatives 3.5.3 Iterated Integrals 3.6 Exercises 117 117 117 121 123 124 128 128 135 138 139 148 156 164 168 168 174 179 180 186 190 201 201 205 209 210 213 217 220 221 222 224 238 246 Points 251 251 251 258 269 277 283 CHAPTER Introduction to Lists and Tables 4.1 Lists and List Operations 4.1.1 Defining Lists 4.1.2 Plotting Lists of Points 4.2 Manipulating Lists: More on Part and Map 4.2.1 More on Graphing Lists: Graphing Lists of Using Graphics Primitives 4.2.2 Miscellaneous List Operations Contents 4.3 Other Applications 4.3.1 Approximating Lists with Functions 4.3.2 Introduction to Fourier Series 4.3.3 The Mandelbrot Set and Julia Sets 4.4 Exercises CHAPTER Matrices and Vectors: Topics from Linear Algebra and Vector Calculus 5.1 Nested Lists: Introduction to Matrices, Vectors, and Matrix Operations 5.1.1 Defining Nested Lists, Matrices, and Vectors 5.1.2 Extracting Elements of Matrices 5.1.3 Basic Computations with Matrices 5.1.4 Basic Computations with Vectors 5.2 Linear Systems of Equations 5.2.1 Calculating Solutions of Linear Systems of Equations 5.2.2 Gauss–Jordan Elimination 5.3 Selected Topics from Linear Algebra 5.3.1 Fundamental Subspaces Associated with Matrices 5.3.2 The Gram–Schmidt Process 5.3.3 Linear Transformations 5.3.4 Eigenvalues and Eigenvectors 5.3.5 Jordan Canonical Form 5.3.6 The QR Method 5.4 Maxima and Minima Using Linear Programming 5.4.1 The Standard Form of a Linear Programming Problem 5.4.2 The Dual Problem 5.5 Selected Topics from Vector Calculus 5.5.1 Vector-Valued Functions 5.5.2 Line Integrals 5.5.3 Surface Integrals 5.5.4 A Note on Nonorientability 5.5.5 More on Tangents, Normals, and Curvature in R 5.6 Matrices and Graphics 5.7 Exercises 283 283 287 299 311 317 317 317 322 325 329 337 337 342 349 349 351 355 358 361 364 366 366 368 374 374 384 387 391 404 415 430 6.1 First-Order Differential Equations 6.1.1 Separable Equations 6.1.2 Linear Equations 435 435 435 442 CHAPTER Applications Related to Ordinary and Partial Differential Equations vii viii Contents 6.2 6.3 6.4 6.5 6.6 6.1.3 Nonlinear Equations 6.1.4 Numerical Methods Second-Order Linear Equations 6.2.1 Basic Theory 6.2.2 Constant Coefficients 6.2.3 Undetermined Coefficients 6.2.4 Variation of Parameters Higher-Order Linear Equations 6.3.1 Basic Theory 6.3.2 Constant Coefficients 6.3.3 Undetermined Coefficients 6.3.4 Laplace Transform Methods 6.3.5 Nonlinear Higher-Order Equations Systems of Equations 6.4.1 Linear Systems 6.4.2 Nonhomogeneous Linear Systems 6.4.3 Nonlinear Systems Some Partial Differential Equations 6.5.1 The One-Dimensional Wave Equation 6.5.2 The Two-Dimensional Wave Equation 6.5.3 Other Partial Differential Equations Exercises 450 453 457 457 458 464 470 472 472 473 475 481 492 492 492 505 511 532 532 537 547 550 References 557 Index 559 Preface Mathematica by Example bridges the gap that exists between the very elementary handbooks available on Mathematica and those reference books written for the advanced Mathematica users This book is an appropriate reference for all users of Mathematica and, in particular, for beginning users such as students, instructors, engineers, businesspeople, and other professionals first learning to use Mathematica This book introduces the very basic commands and includes typical examples of applications of these commands In addition, the text also includes commands useful in areas such as calculus, linear algebra, business mathematics, ordinary and partial differential equations, and graphics In all cases, however, examples follow the introduction of new commands Readers from the most elementary to advanced levels will find that the range of topics covered addresses their needs Taking advantage of Version of Mathematica, Mathematica by Example, Fourth Edition, introduces the fundamental concepts of Mathematica to solve typical problems of interest to students, instructors, and scientists The fourth edition is an extensive revision of the text Features that make this edition easy to use as a reference and as useful as possible for the beginner include the following: Version compatibility All examples illustrated in this book were completed using Version of Mathematica Although many computations can continue to be carried out with earlier versions of Mathematica, we have taken advantage of the new features in Version as much as possible Applications New applications, many of which are documented by references from a variety of fields, especially biology, physics, and engineering, are included throughout the text Detailed table of contents The table of contents includes all chapter, section, and subsection headings Along with the comprehensive index, we hope that users will be able to locate information quickly and easily Additional examples We have considerably expanded the topics throughout the book The results should be more useful to instructors, students, businesspeople, engineers, and other professionals using Mathematica on a variety of platforms In addition, several sections have been added to make it easier for the user to locate information ix 550 CHAPTER Differential Equations 6.6 EXERCISES (a) Solve 1+y y = y cos x (b) Explain the functionality of ProductLog (c) Show that an implicit solution of the equation is 12 y + ln |y| = sin x + C (d) Use ContourPlot to graph various solutions on the rectangle [0, 10] × [0, 10] 2 Solve xy y = y − x and graph several integral curves of the equation (See Figure 6.54(a).) xy xy Solve (−1 + ye + y cos xy) dx + (1 + xe + x cos xy) dy = and graph several integral curves of the equation (See Figure 6.54( b).) Solve y = sin(2x − y), y(0) = 0.5 What is the value of y(1)? Graph for ≤ x ≤ 15 Graph the solution of y = sin(ty), y(0) = j on [0, 7] for j = 0.5, 1, … , 2.5 Create a Manipulate object that lets you compare the solution of x + ax + sin x = to x + ax + x = Solve each of the following differential equations or initial-value problems by hand and then verify your results with Mathematica (a) 2y + 5y + 5y = 0, y(0) = 0, y (0) = 1/2 (b) y + 4y + 13y = t cos 3t a b c d FIGURE 6.54 (a) Integral curves of xy y = y2 − x2 (b) Integral curves of (−1 + yexy + y cos xy) dx + (1 + xexy + x cos xy) dy = (c) The solution of an initial-value problem (d) Solutions to several initial-value problems 6.6 Exercises (c) y − 2y + y = et ln t (d) t y + 16t y + 79ty + 125y = Two lines, l1 and l2 , with slopes m1 and m2 , respectively, are orthogonal (or perpendicular) if their slopes satisfy the relationship m1 = −1/m2 Two curves, C1 and C2 , are orthogonal (or perpendicular) at a point if their respective tangent lines to the curves at that point are perpendicular Now we want to determine the set of orthogonal curves to a given family of curves We refer to this set of orthogonal curves as the family of orthogonal trajectories Suppose that a family of curves is defined as F(x, y) = C and that the slope of the tangent line at any point on these curves is dy/dx = f (x, y) Then, the slope of the tangent line on the orthogonal trajectory is dy/dx = −1/f (x, y) so the family of orthogonal trajectories is found by solving the first-order equation dy/dx = −1/f (x, y) (a) Determine the family of orthogonal trajectories to the family of curves y = cx Confirm your result graphically by graphing members of both families of curves on the same axes (b) Determine the orthogonal trajectories of the family of curves given 2 by y − 2cx = c Graph several members of both families of curves on the same set of axes Why are these two families of curves said to be self-orthogonal? If we are given a family of curves that satisfies the differential equation dy/dx = f (x, y) and we want to find a family of curves that intersects this family at a constant angle , we must solve the differential equation dy f (x, y) ± tan = dx ∓ f (x, y) tan Find a family of curves that intersects the family of curves x2 + y2 = c2 at an angle of /6 Confirm your result graphically by graphing members of both families of curves on the same axes 10 Find a linear differential equation with general solution y = c1 cos t + c2 sin t + et/3 (c3 cos 2t + c4 sin 2t) + 12 t sin t 11 Solve each system and graph various solutions together with the direc0 −1 tion field: (a) X = X, (b) X = X, and (c) x = −1 −1 −5x + 3y, y = −2x − 10y −t 12 Solve x − y = e , y + 5x + 2y = sin 3t, x(0) = x0 , y(0) = y0 Parametrically graph the solution for (x0 , y0 ) = (i, j), where i, j take on four equally spaced values between −1 and 551 552 CHAPTER Differential Equations −␣  X if the eigenvalues of the coefficient matrix are − (a) real and distinct, (b) real and equal, and (c) complex conjugates Hint: Both DSolve and Assumptions might be helpful 14 Under certain assumptions, the FitzHugh–Nagumo equation that arises in the study of the impulses in a nerve fiber can be written as the system of ordinary differential equations 13 Solve X = ⎧ ⎪ dV/d = W ⎪ ⎪ ⎪ ⎨dW/d = F(V) + R − uW ⑀ ⎪ dR/d = (bR − V − a) ⎪ ⎪ u ⎪ ⎩ V(0) = v0 , W(0) = W0 , R(0) = R0 , where F(V) = 13 V − V (a) Graph the solution to the FitzHugh–Nagumo equation that satisfies the initial conditions V(0) = 1, W(0) = 0, and R(0) = if ⑀ = 0.08, a = 0.7, b = 0, and u = (b) Graph the solution that satisfies the initial conditions V(0) = 1, W(0) = 0.5, and R(0) = 0.5 if ⑀ = 0.08, a = 0.7, b = 0.8, and u = 0.6 15 (Controlling the Spread of a Disease) Sources: Herbert W Hethcote, “Three basic epidemiological models,” Applied Mathematical Ecology, edited by Simon A Levin, Thomas G Hallan, and Louis J Gross, Springer-Verlag (1989), pp 119–143; Roy M Anderson and Robert M May, “Directly transmitted infectious diseases: Control by vaccination,” Science, Volume 215, (February 26, 1982), pp 1053–1060; and J D Murray, Mathematical Biology, SpringerVerlag (1990), pp 611–618 If a person becomes immune to a disease after recovering from it and births and deaths in the population are not taken into account, then the percentage (or proportion) of persons susceptible to becoming infected with the disease, S(t), the percentage of people in the population infected with the disease, I(t), and the percentage of the population recovered and immune to the disease, R(t), can be modeled by the system ⎧ S = −SI ⎪ ⎪ ⎪ ⎨I = SI − ␥I ⎪ R = ␥I ⎪ ⎪ ⎩ S(0) = S0 , I(0) = I0 , R(0) = (6.45) Because S(t) + I(t) + R(t) = 1, once we know S(t) and I(t), we can compute R(t) with R(t) = − S(t) − I(t) This model is called an SIR model without vital dynamics because once a person has had the disease, the person becomes immune to the disease, and because births and deaths are not taken into consideration This model might be used to model diseases that are epidemic to a population— those diseases that persist in a population for short periods of time (less than year) Such diseases typically include influenza, measles, rubella, and chickenpox ␥ If S0 < ␥/, I (0) = S0 I0 − ␥I0 < I0 − ␥I0 = Thus, the rate of infection immediately begins to decrease; the disease dies out On 6.6 Exercises the other hand, if S0 > ␥/, I (0) > S0 I0 − ␥I0 > 0, so the rate of infection first increases; an epidemic results Although we cannot find explicit formulas for S, I, and R as functions of t, we can, for example, solve for I in terms of S (S − ␥)I dI =− = −1 + , = ␥/ (a) Solve the equation dS SI S When diseases persist in a population for long periods of time, births and deaths must be taken into consideration If a person becomes immune to a disease after recovering from it and births and deaths in the population are taken into account, then the percentage of persons susceptible to becoming infected with the disease, S(t), and the percentage of people in the population infected with the disease, I(t), can be modeled by the system ⎧ ⎪ ⎨S = −SI + − S I = SI − ␥I − I ⎪ ⎩ S(0) = S0 , I(0) = I0 This model is called an SIR model with vital dynamics because once a person has had the disease, the person becomes immune to the disease, and because births and deaths are taken into consideration This model might be used to model diseases that are endemic to a population—those diseases that persist in a population for long periods of time (10 or 20 years) Smallpox is an example of a disease that was endemic until it was eliminated in 1977 (b) Find and classify the equilibrium points of this system Because S(t) + I(t) + R(t) = 1, it follows that S(t) + I(t) ≤ The following table shows the average infectious period, 1/␥, ␥, and typical contact numbers, , for several diseases during certain epidemics Disease Measles Chickenpox Mumps Scarlet fever 1/␥ 6.5 10.5 19 17.5 ␥ 0.153846 0.0952381 0.0526316 0.0571429 14.9667 11.3 8.1 8.5 Let us assume that the average lifetime, 1/, is 70 years so that = 0.0142857 For each of the diseases listed in the previous table, we use the formula = /(␥ + ) to calculate the daily contact rate 553 554 CHAPTER Differential Equations Disease Measles Chickenpox Mumps Scarlet fever 2.51638 1.23762 0.54203 0.607143 Diseases such as those listed here can be controlled once an effective and inexpensive vaccine has been developed Since it is virtually impossible to vaccinate everybody against a disease, we want to know what percentage of a population needs to be vaccinated to eliminate a disease A population of people has herd immunity to a disease if enough people are immune to the disease so that if it is introduced into the population, it will not spread throughout the population In order to have herd immunity, an infected person must infect less than one uninfected person during the time the person is infectious Thus, we must have S < Since I + S + R = 1, when I = we have that S = − R and, consequently, herd immunity is achieved when (1 − R) < − R < −R < − −1 =1− R> See texts such as Jordan and Smith’s Nonlinear Ordinary Differential Equations [23] for discussions of ways to analyze systems such as the Rossler attractor and the Lorenz equations (c) For each of the diseases listed previously, create a table that estimates the minimum percentage of a population that needs to be vaccinated to achieve herd immunity (d) Using the values in the previous tables, for each disease graph the S = S(t) direction field and several solutions parametrically I = I(t) 16 The Rossler attractor is the system ⎧ ⎪ ⎨x = −y − z y = x + ay ⎪ ⎩ z = bx − cz + xz Observe that this system is nonlinear because of the product of the x and z terms in the z equation 6.6 Exercises If a = 0.4, b = 0.3, x0 = 1, y0 = 0.4, and z(0) = 0.7, how does the value of c affect solutions to the initial-value problem ⎧ x = −y − z ⎪ ⎪ ⎪ ⎨y = x + ay ⎪ z = bx − cz + xz ⎪ ⎪ ⎩ x(0) = x0 , y(0) = y0 , z(0) = z0 ? Suggestion: Use Manipulate 17 Challenge: Using the linear approximation sin = for small displacements, derive the equations for a triple pendulum if theta1 represents the displacement of the upper pendulum (with mass m1 and length l1 ), theta2 represents the displacement of the upper pendulum (with mass m2 and length l2 ), and theta3 represents the displacement of the upper pendulum (with mass m3 and length l3 ) Using g = 32, illustrate the solution graphically if m1 = 3, m2 = 2, and m3 = 1, l1 = 16, l2 = 8, l3 = 16, 1 (0) = 0, 1 (0) = 1, 2 (0) = 0, 2 (0) = 0, 3 (0) = 0, and 3 (0) = −1 555 This page intentionally left blank References [1] Abell, Martha and Braselton, James, Differential Equations with Mathematica, Third Edition, Academic Press, 2004 [2] Abell, Martha and Braselton, James, Modern Differential Equations, Second Edition, Harcourt, 2001 [3] Abell, Martha L., Braselton, James P., and Rafter, John A., Statistics with Mathematica, Academic Press, 1999 [4] Barnsley, Michael, Fractals Everywhere, Second Edition, Morgan Kaufmann, 2000 [5] Braselton, James P., Abell, Martha L., and Braselton, Lorraine M., “When is a surface not orientable?” International Journal of Mathematical Education in Science and Technology, Volume 33, Number 4, 2002, pp 529–541 [6] Devaney, Robert L and Keen, Linda (eds.), Chaos and Fractals: The Mathematics behind the Computer Graphics, Proceedings of Symposia in Applied Mathematics, Volume 39, American Mathematical Society, 1989 [7] Edwards, Henry C and Penney, David E., Calculus with Analytic Geometry, Fifth Edition, Prentice-Hall, 1998 [8] Edwards, Henry C and Penney, David E., Differential Equations and Boundary Value Problems: Computing and Modeling, Third Edition, Pearson/Prentice Hall, 2004 [9] Gaylord, Richard J., Kamin, Samuel N., and Wellin, Paul R., Introduction to Programming with Mathematica, Second Edition, TELOS/Springer-Verlag, 1996 [10] Graff, Karl F., Wave Motion in Elastic Solids, Oxford University Press/Dover, 1975/1991 [11] Gray, Alfred, Abbena, Elsa, and Salamon, Simon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition, CRC Press, 2006 [12] Gray, John W., Mastering Mathematica: Programming Methods and Applications, Second Edition, Academic Press, 1997 [13] Kyreszig, Erwin, Advanced Engineering Mathematics, Seventh Edition, John Wiley & Sons, 1993 [14] Larson, Roland E., Hostetler, Robert P., and Edwards, Bruce H., Calculus with Analytic Geometry, Sixth Edition, Houghton Mifflin, 1998 [15] Maeder, Roman E., The Mathematica Programmer II, Academic Press, 1996 [16] Maeder, Roman E., Programming in Mathematica, Third Edition, Addison-Wesley, 1996 [17] Robinson, Clark, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second Edition, CRC Press, 1999 [18] Smith, Hal L and Waltman, P., The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995 [19] Stewart, James., Calculus: Concepts and Contexts, Second Edition, Brooks/Cole, 2001 557 558 References [20] Weisstein, Eric W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999 [21] Wolfram, Stephen, A New Kind of Science, Wolfram Media, 2002 [22] Zwillinger, Daniel, Handbook of Differential Equations, Second Edition, Academic Press, 1992 Index Symbols /., 102, 121, 149, 185, 376 // N, 109 /;, 119 ;, 46 ?, 17–18, 71–72 ??, 17–18, 71–72 A Absolute convergence, 209 Abs[x], 35–36 AlgebraicManipulation, 37, 44 Alternating harmonic series, 209 Alternating series, 209–210 Amortization, 313–314 Animate Selected Graphics, 131 Annuity due, 312 Antiderivatives, 164–166 Apart, 42–43, 204, 437 Append, 283 AppendTo, 283 Apply, 315 Approximate solutions, 110–114 Arc length, 186–190 Arc length function, 375 ArcCosh[x], 35 ArcCos[x], 35–36 ArcCot[x], 35 ArcCsc[x], 35 ArcCsc[x], 35 ArcSec[x], 35 ArcSinh[x], 35 ArcSin[x], 35–36 ArcTanh[x], 35 Area integrals, 168–174 iterated integrals, 239–240 regions bounded by graph, 180–186 Arithmetic operations, 31–34 Array, 319, 370–371 ArrayPlot, 93–94, 96, 98, 415–416, 420–421, 423–424 Arrow, Arrow, 334–335 AspectRatio, 7, 4, 59, 62, 69, 83, 132, 422 Associated matrix, 358 Assumptions, 41, 552 Astroid, 183–185 A(t), 493 Augmented matrix, 343 Axes, 73–74, 88, 279 AxesLabel, 56, 76–77, 88 AxesOrigin, 73, 142 B Basic Math Input, 5, 9, 11, 32, 135, 203, 224, 273, 318 Beam problem, 158–161 Bendixson’s theorem, 515 Bessel, 295–297 Bessel functions, 274–276, 295, 538–542 BesselJZero, 538–539, 542 Blend, 417 Boxed, 73 BoxRatios, 76, 83, 88–89, 194, 196, 198, 394, 396 Boy surface, 90–92, 116, 431–432 Built-in constants, 34–35 Built-in functions, 35–38 C Cancel[expression], 42–43, 364 Capacitor, 486–487 Catalan, 34, 116 Cell, 10 Cellularautomaton, 95–100 Characteristic equation, 458, 473 Characterstic polynomial, 358–359, 364, 521–522 Circular plate, 294 Clear[f], 117–119, 127, 135, 138, 151, 218, 229, 269, 273, 280, 288 Clothoid, 68 CMYKColor, 417–418 Coefficient matrix, 342 Cofactor matrix, 328 ColorData, 22, 67 ColorFunction, 67, 69, 73–74, 93, 96, 194, 425–426, 429 ColorSchemes, 22, 55, 129, 250, 417, 425 Column, 137, 167 Column space, 349–351 Compile, 546 CompileFunction, 546 ComplexExpand, 484, 489, 491 Compound interest, 246, 311–312 Conditional command, 119 Conditional convergence, 209 Conic section, 81 Conjugate transpose, 364 Conservative vector field, 380–381 Constant coefficients nth-order differential equations, 473–475 second-order differential equations, 458–464 Constants, 34–35 constraints, 198 Continuity, 124–128 Continuum, 126 ContourPlot, 65, 71–73, 76, 79, 104, 108, 111, 113, 116, 139, 222, 227, 229, 231–234, 238, 381–383, 415, 451 ContourPlot3D, 88–89, 105–106 Contours, 73 ContourShading, 73, 111, 142, 451 ContourStyle, 89 Convergence sequence, 201 series, 203–204, 209 tests, 205–207 ConvertTo, 10 Convolution integral, 486 Convolution theorem, 486–487 Cornu spiral, 68 Cosh[x], 35 Cos[x], 35–36 Cot[x], 35 Cramer’s Rule, 471 Create Table/Matrix Palette, 318 Critical number, 148 Critical points, 148, 230–231 Critically damped, 462 Cross product, 331 Cross-Cap, 89 Csc[x], 35 Curl, 382, 389 Curl vector field, 380 Curvature, 375 Cycloid, 143–145 D D, 102, 135–138, 224–225, 229, 267, 381 DarkBands, 425 Deferred annuity, 313 559 560 Index Definite integral, 174–180 Degenerate critical point, 231 Denominator[fraction], 43, 121, 125 DensityPlot, 71–72 Derivatives antiderivatives, 164–166 calculation, 135–138 definition, 128–134 first derivative test, 148–156 mean-value theorem, 146–147 multivariable partial and directional derivatives, 224–233 second derivative test, 148–156 Det, 471 Differential equations first-order differential equations linear differential equations, 442–450 nonlinear differential equations, 450–453 numerical solutions, 453–457 separable differential equations, 435–442 nth-order differential equations constant coefficients, 473–475 Laplace transform, 481–491 nonlinear higher-order equations, 492 theory, 472–473 undetermined coefficients, 475–481 partial differential equations first-order quasilinear partial differential equation, 547–549 one-dimensional wave equation, 532–537 two-dimensional wave equation, 537–547 second-order differential equations constant coefficients, 458–464 theory, 457–458 undetermined coefficients, 464–470 variation of parameters, 470–472 systems of differential equations homogeneous linear systems, 492–505 nonhomogeneous linear systems, 505–532 Differentiation, see also Derivatives antidifferentiation antiderivatives, 164–166 u-substitution, 166–167 implicit differentiation, 138–139 maximization/minimization problems, 156–164 tangent lines, 139–147 Dirac delta function, 490 Direction, 120, 122–124 Direction field, 17 Directional derivative, 225–229 Directory, 14 Disease control, 552 DisplayFunction, 56, 142 Div, 380 Divergence sequence, 201 series, 203–204, 209 test, 205 vector field, 380 Divergence theorem, 388–389 Do, 195, 293, 504, 536, 540, 547 Documentation Center, 2–3, 11, 24–28, 52, 71–72, 95, 339 Dot product, 331 Double pendulum, 500–505, 555 Drawing Tools, 6, 54 Drop, 283 DSolve, 16–17, 435–436, 440–441, 445–446, 448–451, 453, 459–460, 464–465, 471, 474, 476, 478, 481, 489, 495–496, 498–499, 501, 508, 537, 547–548, 552 Dt, 138 Dual problem, 368–369 Duffing’s equation, 614–515 Dynamical system, 264–266, 276, 300–301, 315–316 Dynamic[x], 45, 167 E e, 35 Eigenvalues, 358–361, 366, 430, 518, 521 Eigenvectors, 358–361, 430 Eigenystem, 360–361, 493, 496, 498, 507 Elementary cellular automaton, 95 Ellipse, 81 Ellipsoid, 87 Elliptical torus, 84–86 Endemic disease, 553 Enneper’s minimal surface, 250 Enter, 5, 26, 32 Epidemic, 552 Equation solutions approximate solutions, 110–114 exact solutions, 100–109 Equilibrium point, 514 EulerGamma, 34, 115 Evaluate, 542 Exact differential equation, 450–451 Exact solutions, 100–109 ExampleData, 419 ExpandDenominator[fraction], 43 Expand[expression], 39, 41, 162, 363 ExpandNumerator[fraction], 43, 483 Exponential growth, 445 Expressions algebraic operations, 39–44 defining and evaluating, 47–52 naming and evaluating, 44–46 ExpToTrig, 115 Exp[x], 35–36 F Factor, 459–460, 476 Factor[expression], 39–41, 103, 121, 136, 162, 207, 360–361 Factorial sequence, 202 Falling bodies, 447–448 Family of orthogonal trajectories, 551 Fibonacci numbers, 431 Fibonacci sequence, 311 Filling, 180 Finance, 311–315 FindRoot, 110–116, 193, 219, 248, 296, 449–450 First derivative test, 148–156 First Five Minutes with Mathematica, 25 firstguess, 110 First-order differential equations linear differential equations, 442–450 nonlinear differential equations, 450–453 numerical solutions, 453–457 separable differential equations, 435–442 First-order quasilinear partial differential equation, 547–549 Fit, 283–284, 286–287 FitzHugh–Nagumo equation, 552 Fixed point, 315 Flatten, 126, 262–264, 266, 272, 279, 304, 425, 428, 496 Fobonacci number, 49 Folium of descartes, 375–380 Fourier series defining, 287 kth partial sum, 288 kth term, 287 one-dimensional heat equation, 290–294 partial sums, 288–290 Index wave equation on circular plate, 294–299 Fraction, 46 Frame, 73–74 Frenet formulas, 405 Frenet frame field, 404 FresnelC, 250, 387 FresnelS, 250, 387 FullSimplify, 208, 217 Fundamental matrix, 493 Fundamental set, 457, 473 Fundamental theorem of calculus, 174 Fundamental theorem of line integrals, 384 Future value, 312 fvals, 118 f[x_], 47, 50–51, 147, 169 G Gabriel’s horn, 200–201 Gaussian curvature, 413–414 Gauss–Jordan elimination, 342–349 General form second-order linear differential equation, 457 General solution, 473, 492 Generalized Mandelbrot set, 306–307 Globally asymptotically stable solution, 509 Go, 25 GoldenRatio, 34 Graceful graph, 98 Gradient, 225, 380 GradientFieldPlot3D, 382–383 Gram–Schmidt process, 351–355 Graphics, 6, 54, 131, 278–279, 377 GraphicsArray, 293 GraphicsGrid, 70, 73, 81, 90, 293, 297, 411 Graphics Inspector, 54 GraphicsRow, 86, 88, 130, 139, 278, 280, 304, 334, 421 Graphing cellular automaton, 95–100 functions of single variable, 52–65 parametric and polar plots, 65–70 parametric curves and surfaces in space, 82–94 three-dimensional and contour plots, 71–82 GraphPlot, 98–99 graphs, 81 GrayLevel, 57, 279, 416 Gray’s torus, 84–86 GrayTones, 73–74 Green’s theorem, 385 Grid, 137, 167, 335–337 Growth constant, 445 H Harmonic motion, 461–463 Harmonic series, 207 Hearing beats and resonance, 468–469 Help, 24 Help Browser, 40 Herd immunity, 554 Hermite polynomial, 267–269 Hermitian adjoint matrix, 364 Homogeneous linear differential equation, 442–443 Homogeneous linear systems differential equations, 492–505 Homogeneous nonlinear differential equation, 451 Homogeneous nth-order linear differential equation, 472 Homotopy, 90 Hooke’s Law, 461 Hyperbola, 79, 81 Hyperboloid one sheet, 87 two sheets, 87 I Identity matrix, 321 IE, 34 ihseq, 79 Ikeda map, 304–305 ImageSize, 427 Implicit differentiation, 138–139 Implicit functions tangent lines, 141–142 Import, 92, 419 Indeterminate coordinate, 280 Infinite series, 203 Infinity, 34, 117, 122 -Infinity, 117 Inflection points, 148 Information, 82 Input, 318 InputForm, 10, 38 Insert, 321 Inset, 422 Integrals arc length, 186–190 area, 168–174, 180–186 definite integral, 174–180 iterated integrals, 238–246 solids of revolution, 190–201 vector calculus line integrals, 384–387 surface integrals, 387–391 Integrate, 68, 165–167, 174–179, 181, 183–184, 186–188, 191, 194, 199, 201, 238–239, 242, 244, 267, 291, 375, 383, 385, 389–391, 437, 446, 451–452, 481 Integration by parts formula, 166 Integrating factor, 444 InterpolatingPolynomial, 286 Interval of convergence, 210 Inverse, 325–326, 338 Inverse functions, 58 Inverse Laplace transform, 481, 485 InverseLaplaceTransform, 481, 483, 485, 488, 490, 502–503 Irrotational vector field, 380 Iterated integrals, 238–246 J Jacobian, 514, 520 Join, 328–329, 343 JordanDecomposition, 362–363 Jordan matrix, 361–362 Julia set, 279–282, 299–303, 316 K Kernel linear transformation, 355 Klein bottle orientability, 399–404 Kolmogorov predator–prey equations, 520 L Lagrange multiplier, 235, 237 Lagrange’s equation, 250 Lagrange’s theorem, 236 Laplace transform, 481–491 LaplaceTransform, 481, 483, 487–488, 490, 501–502 Laplacian in polar coordinates, 294 Laplacian of scalar field, 380 leftbox, 168, 170 leftsum, 168–170, 172, 180, 248 Lemniscate of Bernoulli, 185–186 Length, 259, 262, 264, 300, 419, 425, 428 Limit, 46, 117, 119–123, 128–129, 201, 204–206, 208–210 Limit comparison test, 206 Limits computation, 121–123 continuity, 124–128 functions of two variables, 222–224 graphs and tables in prediction, 117–121 one-sided limits, 123–124 561 562 Index Line, 158, 277, 357 Linear differential equations, 442–450 Linear programming dual problem, 368–371 stand form of problem, 366–368 transportation example, 371–374 Linear systems of equations Gauss–Jordan elimination, 342–349 solutions, 337–342 Linear transformations, 355–358 Linearly independent differential equations, 457, 473 LinearProgramming, 369–370 LinearSolve, 339–342 Line integrals, 384–387 List, 101, 141 curve fitting, 283–287 defining, 251–257 graphing, 277–282 manipulation, 269–277 miscellaneous operations, 283 nested list, see Matrix; Vector plotting lists of points, 258–269 list, 118, 135, 254, 259–260, 269, 315, 322 ListContour, 415 ListContourPlot, 422–423, 427–430 ListDensityPlot, 429–430 ListPlot, 61–63, 125–126, 202, 258, 263–264, 266, 278, 280–281, 285–286, 300, 415 ListVectorFieldPlot3D, 333 Locally stable rest point, 514 Logarithmic integral, 261 Logistic equation, 438 Logistic equation with predation, 454–457 Log[x], 35–36 Lorenz equations, 530–532 L–R–C circuit, 486–487 M Mandelbrot set, 299, 305–311 Manipulate, 6–7, 90–92, 132–134, 145, 172–173, 214, 216–217, 248, 251, 257, 335, 378–379, 441, 463, 467, 497–498, 505–506, 510, 513, 519, 524, 529–530, 550, 555 Map, 49, 81, 118, 126, 136–137, 165, 223–224, 248, 267, 270–272, 274–276, 278, 280–282, 304, 308–309, 393, 456, 496, 504, 526 Mathematica information and help resources, 1–2, 17–27 launching, 3–4 package loading, 13–17 syntax rules, 13 user characteristics, version differences, 2–3 MathSource, 13–14 MathWorld, 14, 18, 29, 94 Matrix computations, 325–329 defining, 317–321 element extraction, 322–324 fundamental subspaces, 349–350 graphical representation, 415–430 Jordan matrix, 361–362 Matrix, 318, 321 MatrixForm, 319–320, 323–329, 335, 343–346, 365–366, 480, 521–522 MatrixPlot, 98, 415–417 MatrixPower, 327 Maximization/minimization problems, 156–164 Maximize, 152–156, 193, 247, 369–370 Mclaurin polynomial, 213, 215 Mclaurin series, 213–215 Mean curvature, 413 Mean-value theorem derivatives, 146–147 Mesh, 76–77, 194 MeshFunctions, 65, 83, 196 middlebox, 168, 171, 173 middlesum, 168–170, 172–173, 180, 248 Minimal surface, 249–250 Minimize, 152–156, 193, 247, 367–369, 373 Miscellaneous, 15 ă Mobius strip orientability, 396399 Module, 132 Monotonic sequence, 201 More Information, 19 Multivariable calculus iterated integrals, 238–246 limits of functions of two variables, 222–224 partial and directional derivatives, 224–233 N Names["form"], 21 NDSolve, 433, 440, 453–454, 511, 525, 530 Nest, 60, 276, 300–301 Nested list, see Matrix; Vector Newton’s Second Law, 447–448 NIntegrate, 174, 179–180, 183, 188, 192, 194, 238–239, 378, 543 NMaximize, 193 NMinimize, 193 Nonhomogeneous linear systems differential equations, 505–532 Nonlinear differential equations, 450–453 Nonlinear higher-order differential equations, 492 Norm integral, 174 Normal modes, 294 Normalize, 354–355 Norm[v], 330 NRoots, 110, 113, 183 NSolve, 110, 182 Nullity, 348–349 Nullspace, 347–350, 356 Numerator[fraction], 43, 121, 125 Numerical calculations, 31–34 N[%], 199 N[area], 184 N[expression], 109, 113, 159, 207 N[number], 33, 35–36, 538 O Object, 17 On Line Encyclopedia of Integer Sequences, 201 One-dimensional heat equation, 290–294 One-dimensional wave equation, 532–537 One-sided limits, 123–124 Opacity, 73–74, 86 Options, 17 Options[object], 17–18 Order preserving path, 391 Orientable surface, 391–404 Oriented surface, 388 Orthogonal curves, 145–146 Orthogonal lines, 551 Orthogonalize, 354 Orthonormal vectors, 351 OutputForm, 10 Outward flux vector field, 388–389 Overdamped, 462 Overflow error, 280, 305–306 P Packagename, 14 Packages, 13 Palettes, 11, 22, 44, 318 Panel, 137, 167 Parabola, 81 Parallel vectors, 331 Index Parametric equations arc length, 187 area, 183 tangent lines, 143–145 ParametricPlot, 20–21, 65–67, 69–70, 78, 116, 144–145, 161, 183, 188, 377, 494, 497, 509, 512–513, 516, 522 ParametricPlot3D, 82, 87–88, 90, 191, 196, 198–200, 234, 236–237, 241, 297, 512–513, 546 Part, 254, 322, 418–419 Partial derivative, 224–225 Partial differential equations first-order quasilinear partial differential equation, 547–549 one-dimensional wave equation, 532–537 two-dimensional wave equation, 537–547 Particular solution, 443, 492 Partition, 90, 130, 271–272, 275–276, 293, 429, 504 Pendulum equation with damping, 514 Permutations, 81 , 35 Piecewise, 119 Play, 49 Plot, 5, 14, 21, 52–54, 57, 62, 64, 79, 103, 120, 149–150, 156, 180, 240, 261, 268, 274, 415, 449–450, 472, 477, 489, 512–513 Plot3D, 8, 65, 71, 77, 79, 222, 226, 229, 234, 243, 415, 549 PlotGradientField, 227, 229 PlotJoined, 278, 285 PlotLabel, 56, 69, 73 PlotPoints, 70, 73, 76–77, 88, 112, 200 PlotRange, 56–57, 62, 70, 83, 132, 149, 196, 226, 377, 394, 396 PlotStyle, 7, 14, 55, 57, 67, 69, 73, 86, 144, 195–196, 278, 285 PlotVectorField, 456 Plus, 315 Point, 158, 277, 279, 302 PointSize, 62, 280, 285 PolarPlot, 65–66, 69–70, 78, 116, 185, 189 Potential function, 380 Power series, 210–213 PowerExpand, 41–42, 115, 164, 187, 376 Predator–prey equations, 518–525 Prepend, 283 PrependTo, 283 Present value, 312–13 Prime, 253, 258 Prime number theorem, 261 Principal unit normal vector, 375 Product, 315 Projection vectors, 334, 354–355 Q QEDecomposition, 364–365 Quadric surface, 86–89 Quit, 11 Quit[ ], 11 R RandomInteger, 255 RandomReal, 49, 118, 222–223, 255 Range, 252–253 Rank, 349 Ratio test, 205, 210 RealDigits, 315 RealOnly, 15–16, 28, 34, 63–64, 140, 151, 175–177, 248 Reduce, 213, 217, 522 RegionPlot, 198, 243 RegionPlot3D, 243 Relative maximum, 148, 230–232 Relative minimum, 148, 230–232 ReliefPlot, 93, 415, 427–430 Rendering, 131 ReplaceAll, 46, 106, 121, 149, 185, 376 Rest point, 514 Return, RevolutionPlot3D, 194 RGBColor, 279, 416–417, 424 Right continuous, 124 rightbox, 168, 170 rightsum, 168–170, 172, 174, 180, 248 Roman surface, 90–92 Root test, 206 ă Rossler attractor, 554555 Row space, 349 RowReduce, 329, 343, 346–350 RSolve, 314 S Saddle point, 231–232 Save, 11 seashell, 434 Second derivative test, 148–156, 230 Second-order differential equations constant coefficients, 458–464 theory, 457–458 undetermined coefficients, 464–470 variation of parameters, 470–472 Sec[x], 35 Self-orthogonal curves, 551 Separable differential equations, 435–442 Sequence, 201–202 Series, 215–217 Series alternating series, 209–210 convergence tests, 205–207 harmonic series, 207 infinite series, 203 Mclaurin series, 213–215 power series, 210–213 Taylor series, 213–217 Taylor’s theorem, 217–220 Shading, 76–77 Short, 259, 264, 266, 296, 428 Show, 53–54, 57–58, 63, 70, 73, 86, 88, 92–94, 130, 185, 278–281, 293, 297, 334–335, 357–358, 527 Show Changes, 19 Show More, 420, 423 Sign[x], 64 Simplify, 32, 37–38, 40–42, 59, 128, 136, 146, 164, 213, 223, 229, 242, 375–376, 381, 385, 387, 471, 478, 480–481, 484–485, 488, 491, 508, 522, 534, 535 Sine integral function, 166 Sinh[x], 35 Sin[x], 35 SIR model with vital dynamics, 553 without vital dynamics, 552–553 Slope field, 17 Smooth curve, 187–189 SolarColors, 425 Solids of revolution surface area, 199–201 volume, 190–199 Solve, 100–103, 106, 108–109, 112, 138–139, 147, 149, 154, 157, 207, 237, 240, 338–340, 346–347 Solve, 437, 451, 453, 457, 459–460, 476–477, 487, 490, 515, 520, 522 SolveAlways, 476 SphericalPlot3D, 245–246 Stable fixed point, 315 Standard form first-order linear differential equation, 442 nth-order linear differential equation, 472 second-order linear differential equation, 457 StandardForm, 10–11 Standard unit vectors, 331 Startup Palette, 64 563 564 Index Stayed-wire problem, 163–164 Steady-state temperature, 291 Stokes’ theorem, 389 Sum, 204–205, 207, 209–210 Surface area iterated integrals, 239, 241–242 solids of revolution, 199–201 Surface integrals, 387–391 Surface orientability, 391–404 Syntax rules, 13 Systems of differential equations homogeneous linear systems, 492–505 nonhomogeneous linear systems, 505–532 T Table, 48–49, 52, 60–61, 90, 97, 118, 126, 130, 195–196, 202, 272–273, 214, 222–223, 251–253, 255, 257, 260, 262, 265, 267–268, 279, 292–293, 298, 301, 304, 308–310, 319–320, 323, 485, 496, 516–517, 526, 535, 539, 544–545 TableForm, 268–269, 271–274, 289, 292, 335 TableHeadings, 268–269, 273 Take, 259, 324, 351 Talley, 260 Tangent lines, 139–147 Tangent plane, 233–234 Tanh[x], 35 Tan[x], 35 Taylor polynomial, 213, 215 Taylor series, 213–217 Taylor’s theorem, 217–220 Text, 158, 163–164 Thickness, 86 Thread, 338–339, 348–349 Threadable functions, 136 time, 157 Together[expression], 40–41, 157, 353 Tooltip, 61, 161, 182, 211, 270 Tooth surface, 116 toplot, 61, 81, 144, 219, 300 Torsion, 405 Torus curvature, 414–415 knot, 84–85, 408–409 orientability, 392–398 volume by iterated integral, 245–246 toshow, 293 TraditionalForm, 10, 38–39 Transpose, 323–324, 350, 365 trapezoid, 248 TreeForm, 39 TreePlot, 98–100 Trefoil knot, 411–414 TrigExpand, 37, 103, 115 TrigReduce, 37 TrigToExp, 38 Triple iterated integrals, 244–246 tubeplot, 409 Two-dimensional wave equation, 537–547 U Umbilic torus, 82–83, 432 Underdamped, 462 Underflow error, 280 Undetermined coefficients nth-order differential equations, 475–481 second-order differential equations, 464–470 Union, 96, 526 Unit binormal vector field, 405 Unit circle, 66, 77–79 Unit normal field, 413 Unit normal vector field, 404–405 Unit tangent vector, 375 Unit tangent vector field, 404–407 Unitary matrix, 364 Unstable fixed point, 315 Unstable node, 525 Unstable rest point, 514 Unstable spiral, 525 u-substitution, 166–167 V Van der Pol’s equation, 511, 525–530 Vector calculus line integrals, 384–387 nonorientability, 391–404 surface integrals, 387–391 tangents, normals, and curvature in R3 , 404–415 vector-valued functions, 374–384 computations basic operations, 329–330 projection, 334–337 vectors in 3-space, 330–333 defining, 321–322 Vector triple product, 413 VectorAnalysis, 380, 382, 389 VectorFieldPlot, 381, 438, 494, 516–519 VectorFieldPlots, 15–17, 28, 227, 229–230, 381–383, 393 Verhuist equation, 438 $VersionNumber, VertexLabeling, 100 ViewPoint, 76–77 Volume iterated integrals, 239, 243–244 solids of revolution, 190–199 W Wave equation on circular plate, 294–299 Wronskian, 457, 473, 480 .. .Mathematica by Example This page intentionally left blank Mathematica by Example Fourth Edition Martha L Abell and James P Braselton Department of Mathematical Sciences... encountered by beginning users and are presented in the context of someone familiar with mathematics typically encountered by undergraduates However, for this edition of Mathematica by Example, ... sample, see Examples 2.3.17, 2.3.18, 2.3.21, and 2.3 .23 (b) In Chapter 3, we have improved many examples by adding additional graphics that capitalize on Mathematica s enhanced threedimensional