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Differentiation Rules Constant: Sum: d cϭ0 dx Constant Multiple: d ff(x) Ϯ g(x)g ϭ f 9(x) Ϯ g9(x) dx Quotient: Power: Product: g(x)f 9(x) f(x)g9(x) d f(x) ϭ dx g(x) fg(x)g d n x ϭ nx n dx d cf (x) ϭ c f 9(x) dx d f (x)g(x) ϭ f (x)g9(x) ϩ g(x) f 9(x) dx Chain: d f (g(x)) ϭ f 9(g(x))g9(x) dx Power: d fg(x)g n ϭ nfg(x)g n 1g9(x) dx Derivatives of Functions Trigonometric: 12 d sin x ϭ cos x dx 10 d cos x ϭ Ϫsin x dx 11 d tan x ϭ sec x dx d cot x ϭ Ϫcsc x dx 13 d sec x ϭ sec x tan x dx 14 d csc x ϭ Ϫcsc x cot x dx 17 d tanϪ1 x ϭ dx ϩ x2 d sec Ϫ1 x ϭ dx ZxZ"x 2 20 d csc Ϫ1 x ϭ Ϫ dx ZxZ"x 2 Inverse trigonometric: 15 18 d sinϪ1 x ϭ dx "1 x d cot Ϫ1 x ϭ Ϫ dx ϩ x2 16 19 d cos Ϫ1 x ϭ Ϫ dx "1 x Hyperbolic: 21 d sinh x ϭ cosh x dx 22 d cosh x ϭ sinh x dx 23 d x ϭ sech2 x dx 24 d coth x ϭ Ϫcsch2 x dx 25 d sech x ϭ Ϫsech x x dx 26 d csch x ϭ Ϫcsch x coth x dx 29 d tanhϪ1 x ϭ , ZxZ , dx x2 32 d cschϪ1 x ϭ Ϫ dx ZxZ"x ϩ Inverse hyperbolic: 27 30 d sinhϪ1 x ϭ dx "x ϩ d cothϪ1 x ϭ , ZxZ dx x2 28 31 d coshϪ1 x ϭ dx "x d sechϪ1 x ϭ Ϫ dx x"1 x Exponential: 33 d x e ϭ ex dx 34 d x b ϭ b x(ln b) dx 36 d log b x ϭ dx x(ln b) 38 d dx Logarithmic: 35 d lnZxZ ϭ x dx Of an integral: 37 # x d g(t) dt ϭ g(x) dx a # b a b g(x, t) dt ϭ # 0x g(x, t) dt a Integration Formulas u nϩ1 ϩ C, n Ϫ1 nϩ1 # # u du ϭ lnZuZ ϩ C #e du ϭ e #b du ϭ ln b b #sin u du ϭ Ϫcos u ϩ C #cos u du ϭ sin u ϩ C #sec #csc #sec u tan u du ϭ sec u ϩ C 10 #csc u cot u du ϭ Ϫcsc u ϩ C 11 #tan u du ϭ ϪlnZcos uZ ϩ C 12 #cot u du ϭ lnZsin uZ ϩ C 13 #sec u du ϭ lnZsec u ϩ tan uZ ϩ C 14 #csc u du ϭ lnZcsc u cot uZ ϩ C 15 #u sin u du ϭ sin u u cos u ϩ C 16 #u cos u du ϭ cos u ϩ u sin u ϩ C 17 #sin u du ϭ 18 #cos 19 #sin au sin bu du ϭ 20 #cos au cos bu du ϭ 21 #e 22 #e 23 #sinh u du ϭ cosh u ϩ C 24 #cosh u du ϭ sinh u ϩ C 25 #sech u du ϭ u ϩ C 26 #csch u du ϭ Ϫcoth u ϩ C 27 #tanh u du ϭ ln(cosh u) ϩ C 28 #coth u du ϭ lnZsinh uZ ϩ C 29 #ln u du ϭ u ln u u ϩ C 30 #u ln u du ϭ 2 u ln # "a du ϭ ln P u ϩ "a ϩ u P ϩ C 31 33 35 37 u n du ϭ u u u du ϭ tan u ϩ C au ϩC 2u sin bu du ϭ 14 sin 2u ϩ C sin(a b)u sin(a ϩ b)u ϩC 2(a b) 2(a ϩ b) e au (a sin bu b cos bu) ϩ C a ϩ b2 2 # "a # "a #a 2 du ϭ sin 2 2u 2 u du ϭ u ϩC a u a2 u "a 2 u ϩ sin ϩ C a 2 1 aϩu du ϭ ln P P ϩC a a2u 2u # "u 2 a2 du ϭ ln P u ϩ "u 2 a P ϩ C 32 34 1 u au u ϩC u du ϭ Ϫcot u ϩ C u du ϭ 12 u ϩ 14 sin 2u ϩ C cos bu du ϭ sin(a b)u sin(a ϩ b)u ϩ ϩC 2(a b) 2(a ϩ b) e au (a cos bu ϩ b sin bu) ϩ C a ϩ b2 2 # "a 36 #a 38 # 2 ϩ u2 ϩ u 2du ϭ u 14 u ϩ C u a2 "a ϩ u ϩ ln P u ϩ "a ϩ u P ϩ C 2 1 u du ϭ tanϪ1 ϩ C a a ϩu "u 2 a 2du ϭ u a2 "u 2 a 2 ln P u ϩ "u 2 a P ϩ C 2 ADVANCED SIXTH EDITION Dennis G Zill Loyola Marymount University World Headquarters Jones & Bartlett Learning Wall Street Burlington, MA 01803 978-443-5000 info@jblearning.com www.jblearning.com Jones & Bartlett Learning books and products are available through most bookstores and online booksellers To contact Jones & Bartlett Learning directly, call 800-832-0034, fax 978-443-8000, or visit our website, 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manufacturer, or otherwise does not constitute or imply its endorsement or recommendation by Jones & Bartlett Learning, LLC and such reference shall not be used for advertising or product endorsement purposes All trademarks displayed are the trademarks of the parties noted herein Advanced Engineering Mathematics, Sixth Edition is an independent publication and has not been authorized, sponsored, or otherwise approved by the owners of the trademarks or service marks referenced in this product There may be images in this book that feature models; these models not necessarily endorse, represent, or participate in the activities represented in the images Any screenshots in this product are for educational and instructive purposes only Any individuals and scenarios featured in the case studies throughout this product may be real or fictitious, but are used for instructional purposes only Production Credits VP, Executive Publisher: David D Cella Executive Editor: Matt Kane Acquisitions Editor: Laura Pagluica Associate Editor: Taylor Ferracane Vendor Manager: Sara Kelly Director of Marketing: Andrea DeFronzo VP, Manufacturing and Inventory Control: Therese Connell Composition and Project Management: Aptara®, Inc Cover Design: Kristin E Parker Rights & Media Specialist: Merideth Tumasz Media Development Editor: Shannon Sheehan Cover Images: Domestic: © NASA International: © CHEN MIN CHUN/Shutterstock Printing and Binding: RR Donnelley Cover Printing: RR Donnelley To order this product, use ISBN: 978-1-284-10590-2 Library of Congress Cataloging-in-Publication Data Author: Zill, Dennis G Title: Advanced Engineering Mathematics / Dennis G Zill, Loyola Marymount University Description: Sixth edition | Burlington, MA : Jones & Bartlett Learning, [2017] | Includes index Identifiers: LCCN 2016022410| ISBN 9781284105902 (casebound) | ISBN 1284105903 (casebound) Subjects: LCSH: Engineering mathematics Classification: LCC TA330 Z55 2017 | DDC 620.001/51—dc23 LC record available at https://lccn.loc.gov/2016022410 6048 Printed in the United States of America 20 19 18 17 16 10 Preface Ordinary Differential Equations PART © Andy Zarivny/ShutterStock, Inc Introduction to Differential Equations 1.1 1.2 1.3 Definitions and Terminology Initial-Value Problems Differential Equations as Mathematical Models Chapter in Review First-Order Differential Equations © stefanel/ShutterStock, Inc 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Solution Curves Without a Solution 2.1.1 Direction Fields 2.1.2 Autonomous First-Order DEs Separable Equations Linear Equations Exact Equations Solutions by Substitutions A Numerical Method Linear Models Nonlinear Models Modeling with Systems of First-Order DEs Chapter in Review © ssuaphotos/Shutterstock Contents xiii 14 19 30 33 34 34 36 43 50 59 65 69 74 84 93 99 iii Higher-Order Differential Equations © Tim Jenner/ShutterStock, Inc 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Theory of Linear Equations 3.1.1 Initial-Value and Boundary-Value Problems 3.1.2 Homogeneous Equations 3.1.3 Nonhomogeneous Equations Reduction of Order Homogeneous Linear Equations with Constant Coefficients Undetermined Coefficients Variation of Parameters Cauchy–Euler Equations Nonlinear Equations Linear Models: Initial-Value Problems 3.8.1 Spring/Mass Systems: Free Undamped Motion 3.8.2 Spring/Mass Systems: Free Damped Motion 3.8.3 Spring/Mass Systems: Driven Motion 3.8.4 Series Circuit Analogue Linear Models: Boundary-Value Problems Green’s Functions 3.10.1 Initial-Value Problems 3.10.2 Boundary-Value Problems Nonlinear Models Solving Systems of Linear Equations Chapter in Review The Laplace Transform © azharjggt/ShutterStock, Inc 4.1 4.2 4.3 4.4 4.5 4.6 iv Contents Definition of the Laplace Transform The Inverse Transform and Transforms of Derivatives 4.2.1 Inverse Transforms 4.2.2 Transforms of Derivatives Translation Theorems 4.3.1 Translation on the s-axis 4.3.2 Translation on the t-axis Additional Operational Properties 4.4.1 Derivatives of Transforms 4.4.2 Transforms of Integrals 4.4.3 Transform of a Periodic Function The Dirac Delta Function Systems of Linear Differential Equations Chapter in Review 105 106 106 108 113 117 120 127 136 141 147 151 152 155 158 161 167 177 177 183 187 196 203 211 212 218 218 220 226 226 229 236 237 238 244 248 251 257 Series Solutions of Linear Differential Equations © Cecilia Lim H M/ShutterStock, Inc 5.1 5.2 5.3 Solutions about Ordinary Points 5.1.1 Review of Power Series 5.1.2 Power Series Solutions Solutions about Singular Points Special Functions 5.3.1 Bessel Functions 5.3.2 Legendre Functions Chapter in Review 261 262 262 264 271 280 280 288 294 © sixninepixels/Shutterstock, Inc Numerical Solutions of Ordinary Differential Equations 297 PART 6.1 6.2 6.3 6.4 6.5 Euler Methods and Error Analysis Runge–Kutta Methods Multistep Methods Higher-Order Equations and Systems Second-Order Boundary-Value Problems Chapter in Review Vectors, Matrices, and Vector Calculus © Vaclav Volrab/Shutterstock, Inc Vectors 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Vectors in 2-Space Vectors in 3-Space Dot Product Cross Product Lines and Planes in 3-Space Vector Spaces Gram–Schmidt Orthogonalization Process Chapter in Review 298 302 307 309 313 317 319 321 322 327 332 338 345 351 359 364 Contents v © Image State/Alamy Stock Photo Matrices 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 Matrix Algebra Systems of Linear Algebraic Equations Rank of a Matrix Determinants Properties of Determinants Inverse of a Matrix 8.6.1 Finding the Inverse 8.6.2 Using the Inverse to Solve Systems Cramer’s Rule The Eigenvalue Problem Powers of Matrices Orthogonal Matrices Approximation of Eigenvalues 368 376 389 393 399 405 405 411 415 418 426 430 437 Diagonalization LU-Factorization Cryptography An Error-Correcting Code Method of Least Squares Discrete Compartmental Models Chapter in Review 444 452 459 463 468 472 476 © Dennis Hallinan/Alamy Images Vector Calculus vi Contents 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 367 Vector Functions Motion on a Curve Curvature and Components of Acceleration Partial Derivatives Directional Derivative Tangent Planes and Normal Lines Curl and Divergence Line Integrals Independence of the Path Double Integrals 479 480 486 491 496 501 507 510 516 524 534 Curvilinear motion in the plane, 487 Cycle of a plane autonomous system, 632 Cycloid, 101 Cylindrical coordinates: conversion to rectangular coordinates, 568 definition of, 568 Laplacian in, 755 triple integrals in, 569 Cylindrical functions, 758 Cylindrical wedge, 569 D D’Alembert’s solution, 724 Da Vinci, Leonardo, 19 Damped amplitude, 158 Damped motion, 24, 156, 158–159 Damping constant, 156 Damping factor, 156 Daughter isotope, 97 DE, Decay, radioactive, 21 Decay constant, 74 Decoding a message, 463–464 Definite integral, definition of, 516 Deflation, method of, 441 Deflection curve of a beam, 168 Deflection of a beam, 166–167, 175, 232 Deformation of contours, 869 Degenerate nodes: stable, 639–640 unstable, 639–640 Del operator, 501–502 DeMoivre’s formula, 825 Density-dependent hypothesis, 84 Dependent variables, 496 Derivative of a complex function: of complex exponential function, 840 of complex hyperbolic functions, 847 of complex inverse hyperbolic functions, 850 of complex inverse trigonometric functions, 850 of the complex logarithm function, 844 of complex trigonometric functions, 846 definition of, 833 of integer powers of z, 833 rules for, 833 Derivative of a definite integral, 11, 31 Derivative and integral formulas, APP-2, APP-3 Derivative of a Laplace transform, 237 Derivative of real function, notation for, Derivative of vector function, definition of, 482 Determinant(s): of a 3 matrix, 394 of a matrix, 394 cofactors of, 395 definition of, 393 evaluating by row reduction, 402 expansion by cofactors, 397 of a matrix product, 401 minor of, 395 of order n, 394 of a transpose, 399 of a triangular matrix, 401 properties of, 399 Diagonal matrix, 373, 425 Diagonalizability: criterion for, 446, 448 sufficient condition for, 445, 446 Diagonalizable matrix: definition of, 445 orthogonally, 448 Diagonalization, solution of a linear system of DEs by, 611–612 Difference equation replacement: for heat equation, 807, 809 for Laplace’s equation, 802 for a second-order ODE, 314 for wave equation, 812–813 Difference quotients, 314 Differentiable at a point, 833 Differential: of arc length, 517, 518 of a function of several variables, 59 nth order operator, 108 operator, 108 recurrence relations, 285–286 of surface area, 554 Differential equation (ordinary): Airy’s, 267, 270, 284, 291 associated Legendre’s, 293 autonomous, 36, 150, 646 Bernoulli’s, 67 Bessel’s, 247, 280 Cauchy–Euler, 141 Chebyshev’s, 295 with constant coefficients, 120 definitions and terminology, differential form of, Duffing’s, 193 exact, 59 explicit solution of, families of solutions of, first-order, 6, first-order with homogeneous coefficients, 66 general form of, general solution of, 11, 53, 112, 113, 121–122, 142–144 Gompertz, 87 Hermite’s, 295, 698 higher-order, 123, 139 homogeneous, 51, 66, 108, 120, 142 implicit solution of, Laguerre’s, 248, 697 Legendre’s, 280 linear, 6, 108 as a mathematical model, 19–20 modified Bessel’s, 283 nonhomogeneous, 51, 108, 113 nonlinear, 6, 84, 147, 187 Index INDEX Critical points of an autonomous firstorder differential equation: asymptotically stable, 39 attractor, 39 definition of, 36 isolated, 42 repeller, 39 semi-stable, 39 unstable, 39 Critical points for autonomous linear systems: attractor, 601 center, 640 classifying, 641 definition of, 632 degenerate stable node, 639–640 degenerate unstable node, 639–640 locally stable, 636 repeller, 601 saddle point, 638 stable node, 638 stable spiral point, 640 stability criteria for, 642 unstable, 636 unstable node, 638 unstable spiral point, 640 Critical points for plane autonomous systems: asymptotically stable, 644 classifying, 648 stability criteria for, 647 stable, 644 unstable, 644 Critical speeds, 176 Critically damped electrical circuit, 162 Critically damped spring/mass system, 156 Cross product: component for of, 338 as a determinant, 339 magnitude of, 341 properties of, 339–340 test for parallel vectors, 341 Cross ratio, 925 Crout, Preston D., 458 Crout’s method, 458 Cryptography, 459 Curl of a vector field: definition of, 512 as a matrix product, 375 physical interpretation of, 514, 562 Curvature, 491, 494 Curve integral, 516 Curves: closed, 516 defined by an explicit function, 518 of intersection, 481 parallel, 572 parametric, 480 piecewise smooth, 516 positive direction on, 516 simple closed, 516 smooth, 516 I-5 INDEX I-6 Differential equation (ordinary):—(Cont.) with nonpolynomial coefficients, 269 normal form of, notation for, order of, ordinary, ordinary points of, 264–265 parametric Bessel, 282 parametric modified Bessel, 283 particular solution of, 9, 51, 113 piecewise linear, 54 with polynomial coefficients, 264, 272 Ricatti’s, 69 second-order, 6, 117, 120, 136–137, 141 self-adjoint form of, 695–696 separable, 43–44 singular points of, 264 singular solution of, 10 solution of, 7–8 standard form of a linear, 51, 118, 137 substitutions in, 65 superposition principles for linear, 109, 114 system of, 10, 93, 196, 251 Van der Pol’s, 663, 664 with variable coefficients, 141, 262, 271, 280 Differential equation (partial): classification of linear second-order, 710 definition of, diffusion, 500, 715 heat, 712–713, 716–718, 753 homogeneous linear second-order, 708 Laplace’s, 500, 713, 725 linear second-order, 708 nonhomogeneous linear second-order, 708 order of, Poisson’s, 737 separable, 708–709 solution of, 708 superposition principle for homogeneous linear, 709 time dependent, 732 time independent, 730 wave, 500, 711–712, 719–722, 753 Differential form, 5, 59 Differential operator, nth order, 108 Differential recurrence relation, 285–286 Differentiation of vector functions, rules of, 483 Diffusion equation, 500, 715 Dimension of a vector space, 356 Dirac delta function: definition of, 249 Laplace transform of, 249 Direction angles, 334 Direction cosines, 334 Direction field, 34 Direction numbers of a line, 345 Direction vector of a line, 345 Index Directional derivative: computing, 503 definition of, 502 for functions of three variables, 504 for functions of two variables, 502–503 maximum values of, 504–505 Dirichlet condition, 714 Dirichlet problem: for a circular plate, 748 for a cylinder, 756–757, 764 definition of, 726–727, 803 exterior, 752 harmonic functions and, 918 for a planar region, 803 for a rectangular region, 726 for a semicircular plate, 750 solving using conformal mapping, 919 for a sphere, 760 superposition principle for, 727 Disconnected region, 527 Discontinuous coefficients, 54–55 Discrete compartmental models, 472–473 Discrete Fourier transform, 789 Discrete Fourier transform pair, 790 Discrete signal, 789 Discretization error, 299 Distance formula, 328 Distance from a point to a line, 338 Distributions, theory of, 250 Distributive law: for complex numbers, 821 for matrices, 371 Divergence of a vector field: definition of, 513 physical interpretation of, 514, 578 Divergence theorem, 575 Division of two complex numbers, 820, 824 Domain: in the complex plane, 829 of a complex function, 830 of a function, of a function of two variables, 496 of a solution of an ODE, Dominant eigenvalue, 438 Dominant eigenvector, 438 Doolittle, Myrick H., 454 Doolittle’s method, 454–455 Dot notation for differentiation, Dot product: component form of, 332 definition of, 332, 333 properties of, 332 in terms of matrices, 431 as work, 336 Double cosine series, 743 Double eigenvalues, 750 Double integral: as area of a region, 534, 545 as area of a surface, 553 change of variables, 544 definition of, 534 evaluation of, 536 as an iterated integral, 535 in polar coordinates, 542 properties of, 535 reversing the order of integration in, 537 as volume, 535 Double pendulum, 253 Double sine series, 743 Double spring systems, 155 Doubly connected domain, 859 Downward orientation of a surface, 556 Drag, 24 Drag coefficient, 24 Drag force, 206 Draining a tank, 23, 26 Driven motion: with damping, 158–160 without damping, 160–161 Driving function, 57, 152 Drosophila, 85 Drug dissemination, model for, 82 Drug infusion, 28 Duffing’s differential equation, 193 Dulac negative criterion, 661 Dynamical system, 25, 631 E Ecosystem, states of, 472 Effective spring constant, 155 Effective weight, 490 Eigenfunctions: of a boundary-value problem, 170, of a Sturm-Liouville problem, 693–695 Eigenvalues of a boundary-value problem, 170–171, 692–693 Eigenvalues of a matrix: approximation of, 437 complex, 422, 606 definition of, 418, 599 of a diagonal matrix, 425 distinct-real, 599 dominant, 437–438 of an inverse matrix, 424 of multiplicity m, 602 of multiplicity three, 605 of multiplicity two, 603 repeated, 602 of a singular matrix, 423 of a symmetric matrix, 430, 604 of a triangular matrix, 425 Eigenvector(s) of a matrix: complex, 422 definition of, 418, 599 dominant, 438 of an inverse matrix, 424 orthogonal, 431 Elastic curve, 168 Electrical circuits, 23, 78, 96, 161–163, 241–242 Euler’s constant, 285 Euler’s formula, 121 Euler’s method: error analysis of, 72, 298–301 for first-order differential equations, 71, 298 for second-order differential equations, 309 for systems of differential equations, 312 Evaluation of real integrals by residues, 902–907 Evaporating raindrop, 29, 82 Evaporation, 91 Even function: definition of, 681 properties of, 682 Exact differential: definition of, 59 test for, 59 Exact differential equation: definition of, 59 solution of, 60 Existence and uniqueness of a solution, 16, 106, 594 Existence of Fourier transforms, 783 Existence of Laplace transform, 216 Expansion of a function: in a complex Fourier series, 689 in a cosine series, 683 in a Fourier series, 677–678 in a Fourier–Bessel series, 700 in a Fourier–Legendre series, 702 half-range, 684 in a Laurent series, 887–889 in a power series, 262–263 in a sine series, 683 in terms of orthogonal functions, 674–675 Explicit finite difference method, 808 Explicit solution, Exponential form of a Fourier series, 688 Exponential function: definition of, 840 derivative of, 840 fundamental region of, 841 period of, 841 properties of, 841 Exponential order, 215 Exponents of a singularity, 275 Exterior Dirichlet problem, 752 External force, 158 Extreme displacement, 153 F Falling bodies, mathematical models of, 23–24, 27 Falling chain, 65 Falling raindrops, 29, 82 Family of solutions, Farads (f), 23 Fast Fourier transform, 788, 791 Fast Fourier transform, computing with, 795 Fibonacci, Leonardo, 429 Fibonacci sequence, 429 Fick’s law, 101 Filtered signals, 795 Finite difference approximations, 313–314, 802, 807, 812 Finite difference equation, 315 Finite difference method: explicit, 314, 808 implicit, 809 Finite differences, 314 Finite dimensional vector space, 356 First buckling mode, 171 First harmonic, 722 First moments, 539 First normal mode, 721 First octant, 328 First shifting theorem, 226 First standing wave, 721 First translation theorem: form of, 226 inverse form of, 226 First-order chemical reaction, 21 First-order differential equations: applications of, 74, 64, 93 solution of, 44, 52, 60, 66–68 First-order initial-value problem, 14 First-order Runge–Kutta method, 302 First-order system, 592 Five-point approximation for Laplace’s equation, 802 Flexural rigidity, 168 Flow: around a corner, 939 around a cylinder, 939 of heat, 712 steady-state fluid, 938 Fluctuating population, 82 Flux and Cauchy’s integral formula, 870 Flux through a surface, 556 Folia of Descartes, 13, 652 Forced electrical vibrations, 161 Forced motion: with damping, 158 without damping, 160 Forcing function, 115 Forgetfulness, 28 Formula error, 299 Forward difference, 314 Fossil, age of, 75 Fourier coefficients, 678 Fourier cosine transform: definition of, 783 operational properties of, 784 Fourier integral: complex form, 780–781 conditions for convergence, 778 cosine form, 779 definition of, 777–778 sine form, 779 Fourier integrals, 905 Index INDEX Electrical networks, 96, 252, 255 Electrical vibrations: critically damped, 162 forced, 161–162 free, 162 overdamped, 162 simple harmonic, 162 underdamped, 162 Elementary functions, 11 Elementary matrix, 388 Elementary operations for solving linear systems, 378 Elementary row operations on a matrix: definition of, 380 notation for, 381 Elimination method(s): for a system of algebraic equations, 378–379, 381 for a system of ordinary differential equations, 197 Elliptic partial differential equation, 710 Elliptical helix, 481 Embedded end conditions of a beam, 168, 724 Empirical laws of heat conduction, 712 Encoding a message, 463 Encoding a message in the Hamming (7, 4) code, 464 Entire function, 834 Entries in a matrix, 368 Epidemics, 21, 86, 99 Equality of complex numbers, 820 Equality of matrices, 369 Equality of vectors, 322, 323, 329 Equation of continuity, 578–579 Equation of motion, 153 Equidimensional equation, 141 Equilibrium point, 36 Equilibrium position of a spring/mass system, 152 Equilibrium solution, 36, 632 Equipotential curves, 839 Error(s): absolute, 72 discretization, 299 formula, 299 global truncation, 300 local truncation, 299 percentage relative, 72 relative, 72 round-off, 298 sum of square, 469 Error function, 55, 768 Error-correcting code, 463–467 Error-detecting code, 464, 467 Escape velocity, 194 Essential singularity, 895 Euclidean inner product, 352 Euler, Leonhard, 141 Euler equation, 141 Euler load, 171 Euler–Cauchy equation, 141 I-7 INDEX I-8 Fourier series: complex, 688–690 conditions for convergence, 679 cosine, 683 definition of, 678 generalized, 675 sine, 683 in two variables, 741 Fourier sine transform: definition of, 782 operational properties of, 784 Fourier transform pairs, 783 Fourier transforms: definitions of, 783 existence of, 783 operational properties of, 783–784 Fourier–Bessel series: conditions for convergence, 700 definition of, 698–700 Fourier–Legendre series: conditions for convergence, 702 definition of, 701–702, 703, 704 Fourth-order partial differential equation, 724, 741 Fourth-order Runge–Kutta methods: for first-order differential equations, 72, 303 for second-order differential equations, 309 for systems of differential equations, 311 Free electrical vibrations, 162 Free motion of a spring/mass system: damped, 155–156 undamped, 152 Free vectors, 322 Free-end conditions of a beam, 168, 723 Frequency of free vibrations, 152 Frequency filtering, 795 Frequency response curve, 166 Frequency spectrum, 690 Fresnel sine integral function, 58, 768 Frobenius, Georg Ferdinand, 273 Frobenius, method of, 273 Frobenius’ theorem, 273 Fubini, Guido, 536 Fubini’s theorem, 536 Fulcrum supported ends of a beam, 168 Full-wave rectification of sine, 247 Function(s): complementary, 114 complementary error, 55, of a complex variable, 830 continuous, 832 defined by an integral, 10, 55 differentiable, 833 directional derivative of, 502–503 domain of, 496 driving, 57, 152, 158 error, 55, 768 even, 681 forcing, 115, 152, 158 Fresnel sine integral, 58 Index generalized, 250 gradient of, 502 graph of, 496 harmonic, 515, 837–838, 918 inner product of, 672 integral defined, 10–11 input, 57, 115 odd, 681 orthogonal, 672 output, 57, 115 partial derivative of, 497–498 periodic, 676 polynomial, 832 potential, 575, 937 power, 844, 914 of a real variable, 830 range of, 496 rational, 832 sine integral, 58, 782 stream, 938 of three variables, 497 as a two-dimensional flow, 831 of two variables, 496 vs solution, vector, 480 weight, 675 Fundamental angular frequency, 690 Fundamental critical speed, 176 Fundamental frequency, 722 Fundamental matrix: definition of, 616 matrix exponential as a, 623 Fundamental mode of vibration, 721 Fundamental period, 676, 690 Fundamental region of the complex exponential function, 841 Fundamental set of solutions: definition of, 111, 596 existence of, 112, 596 Fundamental theorem: of algebra, 872–873 of calculus, 11, 526 for contour integrals, 865 for line integrals, 526 G g, 24, 152 Galileo Galilei, 24, 206 Gamma function, 217, 281, APP-4 Gauss’ law, 580, 937 Gauss’ theorem, 575 Gaussian elimination, 381 Gauss–Jordan elimination, 381 Gauss–Seidel iteration, 387, 805 General form of an ordinary differential equation, General solution: of Bessel’s equation, 281, 282 definition of, 11, 53, 112, 113 of a homogeneous linear differential equation, 112 of a homogeneous second-order linear differential equation, 121–122 of a homogeneous system of linear differential equations, 596 of a linear first-order equation, 53 of linear higher-order equations, 123–125 of modified Bessel’s equation, 283 of a nonhomogeneous linear differential equation, 113 of a nonhomogeneous system of linear differential equations, 597 of parametric form of Bessel’s equation, 282 of parametric form of modified Bessel’s equation, 283 of a second-order Cauchy–Euler equation, 142–144 Generalized factorial function, APP-4 Generalized Fourier series, 675 Generalized functions, 250 Generalized length, 673 Geometric series, 879 Geometric vectors, 322 George Washington monument, 168 Gibbs phenomenon, 684 Global truncation error, 300 Globally stable critical point, 659 Gompertz differential equation, 87 Goursat, Edouard, 860 Gradient: of a function of three variables, 501–502 of a function of two variables, 501–502 geometric interpretation of, 507–508 vector field, 511, 525 Gram–Schmidt orthogonalization process, 359–362, 676 Graphs: of a function of two variables, 496 of level curves, 496 of level surfaces, 497 of a plane, 348 Great circles, 572 Green, George, 547 Green’s function: for an initial-value problem, 178 for a boundary-value problem, 184 relationship to Laplace transform, 242–243 for a second-order differential equation, 178 for a second-order differential operator, 178 Green’s identities, 580 Green’s theorem in the plane, 547 Green’s theorem in 3-space, 559 Growth and decay, 21, 74–75 Growth constant, 74 Growth rate, relative, 84 H Half-life: of carbon-14, 75 definition of, 75 Homogeneous systems of linear algebraic equations: definition of, 377, 384 matrix form of, 385 nontrivial solutions of, 384 properties of, 386 trivial solution of, 384 Homogeneous systems of linear differential equations: complex eigenvalues, 606–608 definition of, 592 distinct-real eigenvalues, 599 fundamental set of solutions for, 596 general solution of, 596 matrix form of, 592 repeated eigenvalues, 602–605 superposition principle for, 594 Hoëné-Wronski, Jósef Maria, 111 Hooke’s law, 27, 152 Horizontal component of a vector, 325 Hurricane Hugo, 173–174 Huygens, Christiaan, 206 Hydrogen atoms, distance between, 337–338 Hyperbolic functions, complex: definitions of, 847 derivatives of, 847 zeros of, 848 Hyperbolic partial differential equation, 710 I IC, 14 i, j vectors, 325 i, j, k vectors, 330 Iceman (Ötzi), 101 Identity matrix, 373 Identity property of power series, 263 Ill-conditioned system of equations, 417 Image of a point under a transformation, 581 Images of curves, 912 Imaginary axis, 822 Imaginary part of a complex number, 820 Imaginary unit, 820 Immigration model, 87, 92 Impedance, 163 Implicit finite difference method, 809 Implicit solution, Improper integral: convergent, 212, 904 divergent, 212, 904 Improved Euler method, 300 Impulse response, 250 Incompressible flow, 514, 938 Incompressible fluid, 514 Inconsistent system of linear equations, 377 Indefinite integral, 11, 865 Indented contours, 906 Independence of path: definition of, 526, 864 test for, 527, 528, 529, 864 Independent variables, 496 Indicial equation, 275 Indicial roots, 275 Inductance, 23, 161–162 Infinite-dimensional vector space, 356 Infinite linearly independent set, 357 Infinite series of complex numbers: absolute convergence, 880 definition of, 878 convergence of, 879 geometric, 879 necessary condition for convergence, 879 nth term test for divergence, 880 sum of, 879 Initial conditions (IC), 14, 106, 714 Initial-value problem (IVP): definition of, 14, 106 first-order, 14, 53 nth-order, 14, 106 second-order, 14 for systems of linear differential equations, 594 Inner partition, 564 Inner product: of two column matrices, 431 definition of, 332, 333, 672 properties of, 347, 672 space, 358 of two functions, 672 of two vectors, 332, 672 Inner product space, 358 Input function, 57, 115 Insulated boundary, 714 Integers: modulo 2, 463 modulo 27, 462 Integrable function: of three variables, 565 of two variables, 534 Integral-defined function, 10–11 Integral equation, 241 Integral transform: definition of, 212 Fourier, 783 Fourier cosine, 783 Fourier sine, 783 inverse, 782, kernel of, 782 Laplace, 212, 782 pair, 782 Integral of a vector function, 484 Integrating factor, 52, 62–63 Integration along a curve, 516 Integration by parts, 877 Integrodifferential equation, 241 Interest, compounded continuously, 80 Interior point, 802 Interior mesh points, 314 Interior point of a set in the complex plane, 828 Interpolating function, 306 Index INDEX of a drug, 21 of plutonium-239, 75 of radium-226, 75 of uranium-238, 75 Half-plane, 828 Half-range expansions, 685 Half-wave rectification of sine, 247 Hamilton, William Rowan, 351 Hamming (7, 4) code, 464 Hamming (8, 4) code, 468 Hamming, Richard W., 464 Hard spring, 188 Harmonic conjugate functions, 838 Harmonic function, 515, 837–838, 918 Harmonic function, transformation theorem for, 918 Harmonic functions and the Dirichlet problem, 918 Harvesting, 86 Heart pacemaker, model for, 58, 83 Heat equation: derivation of one-dimensional equation, 712 difference equation replacement for, 807, 809 and discrete Fourier series, 791 and discrete Fourier transform, 791–792 one-dimensional, 711–712 in polar coordinates, 753 solution of, 716 two-dimensional, 753 Heaviside, Oliver, 229 Heaviside function, 229 Helmholtz’s partial differential equation, 763 Helix: circular, 480 elliptical, 481 pitch of, 481 Henrys (h), 23 Hermite, Charles, 295 Hermite polynomials, 295 Hermite’s differential equation, 295, 698 Higher-order ordinary differential equations, 105, 123, 141 Hinged end of a beam, 168 Hitting bottom, 92 Hole through the Earth, 28 Homogeneous boundary conditions, 169, 183, 693 Homogeneous boundary-value problem, 169, 715 Homogeneous first-order differential equation: definition of, 66 solution of, 66 Homogeneous function, 66 Homogeneous linear differential equation: ordinary, 51, 108 partial, 708 I-9 INDEX Interval: of convergence, 262 of definition of a solution, of existence and uniqueness, 16 of validity of solution, Invariant region: definition of, 662 Types I and II, 662 Invasion of the marine toads, 103 Inverse cosine function: derivative of, 850 as a logarithm, 849 Inverse hyperbolic functions: definition of, 850 derivatives of, 850 as logarithms, 850 Inverse integral transform: Fourier, 783 Fourier cosine, 783 Fourier sine, 783 Laplace, 218, 782 Inverse of a matrix: definition of, 405 by the adjoint method, 406–407 by elementary row operations, 409 properties of, 406 using to solve a system, 411–412 Inverse power method, 443 Inverse sine function: definition of, 849 derivative of, 850 as a logarithm, 849 Inverse tangent function: derivative of, 850 as a logarithm, 849 Inverse transform, 218, 782, 783 Inverse transformation, 582 Inverse trigonometric functions: definitions of, 849 derivatives of, 850 Invertible matrix, 405 Irregular singular point, 272 Irrotational flow, 514, 938 Isocline, 35, 41 Isolated critical point, 42 Isolated singularity: classification of, 894–895 definition of, 887 Iterated integral, 535 IVP, 14 J Jacobian determinant, 582 Jacobian matrix, 647 Joukowski airfoil, 915 Joukowski transformation, 915 K Kepler’s first law of planetary motion, 490 Kernel of an integral transform, 782 Kinetic friction, 54 Kirchhoff’s first law, 96 I-10 Index Kirchhoff’s point and loop rules, 383 Kirchhoff’s second law, 23, 96 L Lagrange’s identity, 344 Laguerre polynomials, 248 Laguerre’s differential equation, 248, 697 Laplace, Pierre-Simon Marquis de, 213 Laplace transform: behavior as s S q, 223 of Bessel function of order n ϭ 0, 247 conditions for existence, 216 convolution theorem for, 238–239 definition of, 212, 782 derivatives of a, 237 of derivatives, 220 of differential equations, 221 differentiation of, 237 of Dirac delta function, 249 existence of, 216 inverse of, 218, 782 of an integral, 240 as a linear transform, 214 and the matrix exponential, 623 of a partial derivative, 770 of a periodic function, 244 of systems of ordinary differential equations, 251 tables of, 215, APP-6 translation theorems for, 226, 230 of unit step function, 230 Laplace’s equation, 486, 500, 501, 711–712, 725, 748, 802 Laplace’s partial differential equation: in cylindrical coordinates, 756 difference equation replacement for, 802 in polar coordinates, 748 maximum principle for, 727 solution of, 725 in three dimensions, 744 in two dimensions, 711–712, 725 Laplacian: in cylindrical coordinates, 755 definition of, 515, 712 in polar coordinates, 748 in rectangular coordinates, 712 in spherical coordinates, 760 in three dimensions, 712 in two dimensions, 712 Lascaux cave paintings, dating of, 80 Latitude, 572 Lattice points, 802 Laurent series, 888 Laurent’s theorem, 889–890 Law of conservation of mechanical energy, 533 Law of mass action, 87 Law of universal gravitation, 28 Laws of exponents for complex numbers, 845 Laws of heat conduction, 712 Leaking tank, 89–90 Leaning Tower of Pisa, 24 Learning theory, 28 Least squares, method of, 468–470 Least squares line, 92, 469 Least squares parabola, 470–471 Least squares solution, 470 Legendre, Adrien-Marie, 280 Legendre associated functions, 293 Legendre functions, 288–289, 293, 294–295 Legendre polynomials: first six, 289 graphs of, 289 properties of, 289 recurrence relation for, 289 Rodriques’ formula for, 290 Legendre’s differential equation: associated, 293 of order n, 280 series solution of, 288–289 Leibniz notation, Leibniz’s rule, 31 Length of a space curve, 484 Length of a vector: in 3-space, 333 in n-space, 352 Leonardo da Vinci, 19 Level curves, 46, 496 Level of resolution of a mathematical model, 19 Level surfaces, 497 L’Hôpital’s rule, 160–161, 216, 282, 901 Liber Abbaci, 429 Libby half-life, 75 Libby, Willard F., 75 Liebman’s method, 806 Limit cycle, 662, 664 Limit of a function of a complex variable: definition of, 832 properties of, 832 Limit of a vector function, 481 Line of best fit, 469 Line integrals: around closed paths, 519, 528, 546–547 as circulation, 522 complex, 854 in the complex plane, 854 definition of, 516–517 evaluation of, 517, 518, 520 fundamental theorem for, 526 independent of the path, 526 in the plane, 517 in space, 520 as work, 521 Line segment, 345 Lineal element, 34 Linear algebraic equations, systems of, 376–377 Linear combination of vectors, 324 nonhomogeneous, 708 solution of, 708 superposition principle for, 709 Linear spring, 188 Linear system: of algebraic equations, 376 definition of, 377 of differential equations, 93, 592 rank and, 392 Linear transform, 214, 219 Linearity: of a differential operator, 108 of the inverse Laplace transform, 219 of the Laplace transform, 212 Linearity property, 108 Linearization: of a function f(x) at a number, 70, 643 of a function f(x, y) at a point, 643 of a nonlinear differential equation, 189 of a nonlinear system of differential equations, 645–646 Linearly dependent set of functions, 109 Linearly independent set of functions, 109 Lines of force, 511 Lines in space: direction numbers for, 345 direction vector for, 345 normal, 509 parametric equations of, 345 symmetric equations of, 346 vector equation for, 345 Liouville’s theorem, 872 Lissajous curve, 202, 256 Local linear approximation, 70, 643, 646 Local truncation error, 299 Locally stable critical point, 636 Logarithm of a complex number: branch cut for, 844 branch of, 843 definition of, derivative of, 844 principal branch, 843 principal value of, 843 properties of, 844 Logistic curve, 85 Logistic equation: definition of, 69, 84–85 modifications of, 86 solution of, 85 Logistic function, 85 Logistic growth, 84–86 Longitude, 572 Loop rule, Kirchhoff’s, 383 Losing a solution, 45 Lotka–Volterra competition model, 96 Lotka–Volterra predator-prey model, 95 Lower bound for the radius of convergence, 265 Lower triangular matrix, 372 LRC-series circuit: differential equation of, 23, 161–162 integrodifferential equation of, 241 LR-series circuit, differential equation of, 78 LU-decomposition of a matrix, 452 LU-factorization of a matrix, 452–455 M Maclaurin series, 263, 884, 885 Maclaurin series representation: for the cosine function, 263 for the exponential function, 263 for the sine function, 263 Magnification in the z-plane, 913 Magnitude of a complex number, 430 Magnitude of the cross product, 341, 342 Magnitude of a vector, 324, 333 Main diagonal entries of a matrix, 368 Malthus, Thomas, 20 Malthusian model, 20 Mapping, 581, 912 Mapping, conformal, 916 Marine toad invasion model, 103 Mass: center of, 538 as a double integral, 538 of a surface, 554 Mass action, law of, 87 Mathematical model, 19–20, 151, 167, 187 Matrix (matrices): addition of, 369 adjoint, 406 associative law, 370, 371 augmented, 380 banded, 804 characteristic equation of, 419 coefficient, 385 column vector, 368 commutative law, 370 definition of, 368 determinant of, 393–395 diagonal, 373, 425 diagonalizable, 445 difference of, 369 distributive law, 371 dominant eigenvalue of, 437–438 eigenvalues of, 418, 422, 424, 425 eigenvectors of, 418, 422, 424, 425 elementary, 388 elementary row operations on, 380–381 entries (or elements) of, 368 equality of, 369 exponential, 612–622 fundamental, 616 identity, 373 inverse of, 405, 407–408, 409 invertible, 405 Jacobian, 647 lower triangular, 372, 425 LU-factorization of, 452–453 main diagonal entries of, 368 multiplication, 370 multiplicative identity, 373 Index INDEX Linear dependence: of a set of functions, 109–110 of a set of vectors, 355, 357 of solution vectors, 595 Linear donor-controlled hypothesis, 472 Linear equation in n variables, 376 Linear first-order differential equation: definition of, 6, 50 general solution of, 53 homogeneous, 51 integrating factor for, 52 method of solution, 52 nonhomogeneous, 51 singular points of, 53 standard form of, 51 variation of parameters for, 51 Linear fractional transformation, 923 Linear independence: of a set of functions, 109–110 of a set of vectors, 355 of solution vectors, 595 of solutions of linear DEs, 110–111 Linear momentum, 490 Linear operator, 108, 197 Linear ordinary differential equations: applications of, 58, 74–84, 151, 167 associated homogeneous, 108 auxiliary equation for, 120, 142 boundary-value problems for, 107, 167, 183 with constant coefficients, 120 complementary function for, 114 definition, 6, first-order, 6, 50 general solution of, 53, 112, 113, 121–122, 142–143 higher-order, 106, 108, 123, 141 homogeneous, 51, 108, 120 indicial equation for, 275 initial-value problems for, 14, 74, 106, 149, 151, 177 infinite series solutions for, 265, 273 nonhomogeneous, 51, 127, 136 nth-order initial-value problem for, 106 ordinary points of, 264–265 particular solution for, 51, 113, 127, 136 piecewise, 54 reduction of order, 117–119 second-order, 6, 107, 117–119, 120 singular points of, 53, 264–265, 272, 273 standard forms of, 51, 118, 137, 139, 177 superposition principles for, 109, 114 with variable coefficients, 141, 261, 264, 269, 271–278, 280 Linear partial differential equation, 708 Linear regression, 103 Linear second-order partial differential equations: classification of, 710 homogeneous, 708 I-11 INDEX I-12 Matrix (matrices):—(Cont.) multiplicative inverse, 405 nilpotent, 430, 476, 626 null-space of, 387 nonsingular, 405, 406 order n, 368 orthogonal, 413, 433–434 orthogonally diagonalizable, 448 partitioned, 376 powers of, 426 product of, 370 rank of, 389–390 reduced row-echelon form, 381 rotation, 375 row-echelon form, 381 row equivalent, 381 row reduction of, 381 row space of, 389 row vector, 368 scalar, 373 scalar multiple of, 369 similar, 426 singular, 406 size, 368 skew-symmetric, 405 sparse, 804 square, 368 stochastic, 426 sum of, 369 symmetric, 373, 604 of a system, 380 trace of a, 636 transpose of, 371 triangular, 372, 425 tridiagonal, 810 upper triangular, 372 zero, 372 Matrix addition, properties of, 370 Matrix exponential: computation of, 622, 623 definition of, 622 derivative of, 623 as a fundamental matrix, 623 as an inverse Laplace transform, 623 Matrix form of a system of linear algebraic equations, 385 Matrix form of a system of linear differential equations, 592 Maximum principle, 727 Maxwell, James Clerk, 514 Maxwell’s equations, 515 Meander function, 247 Memorization, mathematical model for, 28 Meridian, 572 Mesh: points, 314, 802 size, 802 Message, 463 Methane molecule, 337–338 Method of deflation, 441 Method of diagonalization: for homogeneous systems of linear DEs, 511 Index for nonhomogeneous systems of linear DEs, 619 Method of Frobenius, 273–277 Method of isoclines, 35 Method of least squares, 468–470 Method of separation of variables: for ordinary differential equations, 43–44 for partial differential equations, 708 Method of undetermined coefficients: for nonhomogeneous linear DEs, 127 for nonhomogeneous systems of linear DEs, 614–616 Method of variation of parameters: for nonhomogeneous linear DEs, 51, 136–137 for nonhomogeneous systems of linear DEs, 616–619 Midpoint of a line segment in space, 329 Minor determinant, 395 Mises, Richard von, 438 Mixed boundary conditions, 693 Mixed partial derivatives: definition of, 498 equality of, 498 Mixtures, 22, 77, 94 ML-inequality, 857 Mm, n vector space, 373 Möbius strip, 555 Modeling process, steps in, 20 Modifications of the logistic equation, 86–87 Modified Bessel equation: of order n, 283 parametric form of, 283 Modified Bessel function: of the first kind, 283 of the second kind, 283 Modulus of a complex number, 822 Moments of inertia, 539 Moments of inertia, polar, 546 Motion: on a curve, 486 in a force field, 151 Moving trihedral, 492 Multiplication: of complex numbers, 820, 824 of matrices, 370 of power series, 263 by scalars, 322, 323, 329 Multiplication rule for undetermined coefficients, 132 Multiplicative inverse of a matrix, 405 Multiplicity of eigenvalues, 421, 602–605 Multiply connected domain, 859 Multiply connected region, 527 Multistep numerical method, 307 N n-dimensional vector, 351–352 Negative criteria, 660, 661 Negative direction on a curve, 547 Negative of a vector, 323 Neighborhood, 828 Net flux, 512 Networks, 96, 252 Neumann condition, 714 Neumann problem: for a circular plate, 753 for a rectangle, 729 Newton, Isaac, 206 Newton’s dot notation, Newton’s law of air resistance, 206 Newton’s law of cooling/warming, 21, 76 Newton’s law of universal gravitation, 28 Newton’s laws of motion: first, 23 second, 23, 27, 191–192 Nilpotent matrix, 430, 476, 626 Nodal line, 755 Nodes: of a plane autonomous system, 638–640, 642 of a standing wave, 721 Nonconservative force, 533 Nonelementary integral, 11, 55 Nonhomogeneous boundary condition, 693 Nonhomogeneous boundary-value problem, 169, 183, 693, 730 Nonhomogeneous linear differential equation: definition of, 108 general solution of, 113 initial-value problem for, 106 ordinary, 51, 108 partial, 708 particular solution of, 113 Nonhomogeneous systems of algebraic equations, 377 Nonhomogeneous systems of linear differential equations: complementary function of, 597 definition of, 592 general solution of, 597 initial-value problem for, 594 matrix form of, 592 normal form of, 592 particular solution of, 596 solution vector of, 593 Nonisolated singular point, 888 Nonlinear mathematical models, 84, 652 Nonlinear ordinary differential equation, 6, 147 Nonlinear oscillations, 653 Nonlinear pendulum, 189, 652–653 Nonlinear spring, 188 Nonlinear systems of differential equations, 93, 629 Nonoriented surface, 555 Nonpolynomial coefficients, 269 Nonsingular matrix, 405, 406, 409 Nontrivial solution, 169 predictor-corrector methods, 300, 307 Runge–Kutta methods, 72, 302, 309, 311 shooting method, 316 single-step method, 307 stability of, 308 stable, 308 starting method, 307 unstable, 308 using the tangent line, 71 Numerical solution curve, 73 Numerical solver, 72 Numerical values of Bessel functions, 285 O Octants, 328 Odd function: definition of, 681 properties of, 682 ODE, Ohms (⍀), 23 Ohm’s law, 79 One-dimensional heat equation: definition of, 711–712 derivation of, 712–713 One-dimensional phase portrait, 37 One-dimensional wave equation: definition of, 711–712 derivation of, 713 One-parameter family of solutions, One-to-one transformation, 582 Open annulus, 829 Open disk, 828 Open region, 527 Open set, 828 Operational properties of the Laplace transform, 214, 220, 226, 230, 231, 237, 239, 240, 244, 249 Operator, differential, 108, 197 Order of a differential equation, 4, Order, exponential, 215 Order of a Runge–Kutta method, 302–303 Order of integration, 537, 565–567 Ordered n-tuple, 351–352, 354 Ordered pair, 323, 327, 351, 354 Ordered triple, 328, 351, 354 Ordinary differential equation, Ordinary point of an ordinary differential equation: definition of, 264, 265 solution about, 265–269 Orientable surface: definition of, 555 of a closed, 556 Orientation of a surface: downward, 556 inward, 556 outward, 556 upward, 556 Orthogonal basis for a vector space, 359, 360, 361, 362 Orthogonal diagonalizability: criterion for, 448 definition of, 448 Orthogonal eigenvectors, 431–432 Orthogonal family of curves, 102, 839 Orthogonal functions, 672 Orthogonal matrix: constructing an, 434 definition of, 433 Orthogonal projection of a vector onto a subspace, 361 Orthogonal series expansion, 674–675 Orthogonal set of functions, 673 Orthogonal with respect to a weight function, 675 Orthogonal surfaces at a point, 510 Orthogonal trajectories, 102 Orthogonal vectors, 333 Orthogonally diagonalizable matrix, 448 Orthonormal basis: definition of, 359 for R n, 359 for a vector space, 359 Orthonormal set of functions, 673 Orthonormal set of vectors, 433 Oscillating chain, 759 Osculating plane, 492 Ötzi (the iceman), 101 Output function, 57, 115 Overdamped electrical circuit, 162 Overdamped spring/mass system, 156 Overdamped system, 654 Overdetermined system of linear algebraic equations, 386 Overtones, 722 INDEX Norm: of a column vector (matrix), 431 of a function, 673 of a partition, 516, 534 square, 673 of a vector, 324, 352, 358 Normal component of acceleration, 492 Normal form: of an ordinary differential equation, of a system of linear first-order equations, 592 Normal line to a surface, 509 Normal modes, 721 Normal plane, 492 Normal vector to a plane, 347 Normalization of a vector, 324, 352 Normalized eigenvector, 434 Normalized set of orthogonal functions, 674 Notation for derivatives, n-parameter family of solutions, n-space (R n): coordinates relative to an orthonormal basis, 356 dot (or inner product) in, 352 length (or norm) in, 352 orthogonal vectors in, 352 orthonormal basis for, 359 standard basis for, 356 unit vector in, 352 vector in, 352 zero vector in, 352 nth root of a nonzero complex number, 825–826 nth roots of unity, 797 nth term test for divergence, 880 nth-order differential equation expressed as a system, 310 nth-order differential operator, 108 nth-order initial-value problem, 14, 106 nth-order ordinary differential equation, 5–6, 106, 108 Nullcline, 42 Null-space of a matrix, 387 Number of parameters in a solution of a linear system of equations, 391 Numerical methods: absolute error in, 72 Adams–Bashforth–Moulton, 307 adaptive methods, 305 continuing method, 307 Crank–Nicholson method, 809 deflation method, 441 errors in, 72, 298 Euler’s method, 71, 298–300, 305, 309, 312 finite-difference methods, 314, 802, 807, 812 Gauss–Seidel iteration, 805 improved Euler’s method, 300 inverse power method, 443 multistep method, 307 power method, 438 P Pacemaker, heart, 58, 83 Parabolic partial differential equation, 710 Parallel vectors: definition of, 322 criterion for, 341 Parallels, 572 Parametric curve: closed, 516 definition of, 480 piecewise smooth, 516 positive direction on, 516 simple closed, 516 smooth, 482, 516 in space, 480 Parametric equations for a line in space, 345 Parametric form of Bessel equation: of order n, 696 of order n, 282 in self-adjoint form, 696 Parametric form of modified Bessel equation of order n, 283 Parent isotope, 97 Paris Guns, 206–209 Parity, 463 Index I-13 INDEX I-14 Parity check bits, 463 Parity check code, 463 Parity check equations, 465 Parity check matrix, 465 Parity error, 464 Partial derivatives: Chain Rule for, 498–499 definition of, 497 generalizations of, 499 higher-order, 498 mixed, 498 with respect to x, 497 with respect to y, 497 second-order, 498 symbols for, 498 third-order, 498 tree diagrams for, 499 Partial differential equation, linear second order: definition of, 708 elliptic, 710, 802 homogeneous, 708 hyperbolic, 710, 802, 812 linear, 708 nonhomogeneous, 708 parabolic, 710, 802, 807 separable, 708 solution of, 708 Partial fractions, use of, 219, 223–224 Particular solution: definition of, 9, 113 of Legendre’s equation, 288–289 of a nonhomogeneous system of linear DEs, 596, 614, 616, 619 by undetermined coefficients, 127–134 by variation of parameters, 136–140 Partitioned matrix, 376 Path independence: definition of, 526 tests for, 529, 531 Path of integration, 525 Pauli spin matrices, 476 PDE, Pendulum: ballistic, 205 double, 253–254 free damped, 191 linear, 189 nonlinear, 189 oscillating, 190 physical, 189 rotating, 668 simple, 187–188 spring, 205–206 spring-coupled, 259 of varying length, 292–293 whirling, 190 Percentage relative error, 72 Perihelion, 491 Period: of the complex exponential function, 841 of the complex sine and cosine, 848 Index of the complex hyperbolic sine and cosine, 848 Period of free vibrations, 152 Periodic boundary conditions, 176, 695 Periodic boundary-value problem, 695 Periodic driving force, 160, 686 Periodic extension, 680 Periodic functions: definition of, 244, 676, 841 Laplace transform of, 244 Periodic solution of a plane autonomous system, 632 Phase angle, 153 Phase line, 37 Phase plane, 593, 600, 637 Phase portrait: for first-order differential equations, 37 for systems of two linear first-order differential equations, 600, 637 for systems of two nonlinear firstorder differential equations, 648–650 Phase-plane method, 649 Physical pendulum, 189 Piecewise-continuous function: definition of, 215 Laplace transform of, 216 Piecewise-defined solution of an ordinary differential equation, 10, 47 Piecewise-linear differential equation, 54, 211 Piecewise-smooth curve, 516 Pin supported end of a beam, 168 Pitch, 375–376 Pitch of a helix, 481 Planar transformation, 912 Plane(s): Cartesian equation of, 347 curvilinear motion in, 487 graphs of, 348–349 line of intersection of two, 349 normal vector to, 347 perpendicular to a vector, 347–348 phase, 600, 637 point-normal form of, 347 trace of, 348 vector equation of, 347 Plane autonomous system: changed to polar coordinates, 633 types of solutions of a, 632 in two variables, 631 Plane autonomous system, solutions of: arc, 632 constant, 632 periodic (cycle), 632 Plucked string, 714, 720–721, 725 Plutonium-239, half-life of, 75 Poincare–Bendixson theorems, 662, 664–665 Point-normal form of an equation of a plane, 347 Point rule, Kirchhoff’s, 383 Poisson integral formula: for unit disk, 934 for upper half-plane, 932–933 Poisson’s partial differential equation, 737 Polar coordinates, 544–545, Polar form of a complex number, 823–824 Polar moment of inertia, 546 Polar rectangle, 542 Pole: definition of, 894–895 of order n, 894–895, 896 residue of, 897–898 simple, 894–895 Polynomial function, 832 Population, mathematical models for, 20, 69, 74, 84, 86–87, 94–96, 654, 656 Position vector, 323, 329 Positive criteria, 662 Positive direction on a curve, 516, 546–547 Potassium-argon dating, 76, 97 Potassium-40 decay, 97 Potential: complex, 937 complex velocity, 938 energy, 534 function, 525, 937 Power function, 914 Power method, 438 Power rule of differentiation, 833, APP-2 Power series: absolute convergence of, 262 arithmetic of, 263 center of, 262 circle of convergence, 881 convergence of, 262 defines a function, 263 definition of, 262 differentiation of, 263 identity property of, 263 integration of, 263 interval of convergence, 262 Maclaurin, 263 radius of convergence, 262 ratio test for, 262 represents a continuous function, 263 represents an analytic function, 263 review of, 262–263 shift of summation index, 263 solutions of differential equations, 264–265 Taylor, 263 Powers, complex, 844 Powers, integer, 825 Powers of a matrix, 426–428 Predator-prey model, 94–95, 654–655 Predictor-corrector methods, 300, 307 Prime meridian, 572 Prime notation, Principal argument of a complex number, 824 Principal axes of a conic, 451 Principal branch of the logarithm, 843 Principal logarithmic function, 843 Principal normal vector, 492 Principal nth root of a complex number, 826 Principal part of a Laurent series, 888, 894 Principal value: of a complex power, 844 of an integral, 904 of logarithmic function, 843 Product Rule, 833, APP-2 Projectile motion, 202, 206–209, 255–256 Projection of a vector onto another, 335 p-series, 880 Pure imaginary number, 820 Pure resonance, 160–161 Pursuit curve, 194–195 Pythagorean theorem, 324 Q R Radial symmetry, 753 Radial vibrations, 753 Radioactive decay, 21, 74–75 Radioactive decay series, 93 Radiogenic isotope, 97 Radiometric dating methods, 76 Radius of convergence, 262, 881 Radius of curvature, 495 Radius of gyration, 540 Raindrop, velocity of evaporating, 82 Raleigh differential equation, 651 Range of a complex function, 830 Range of a projectile: with air resistance, 256 with no air resistance, 255 Rank of a matrix: definition of, 389 by row reduction, 389–390 Ratio test, 262, 880 Rational function, 832 Rational roots of a polynomial equation, 124 Rayleigh quotient, 439 RC-series circuit, differential equation of, 78 zero-input, 224 zero-state, 224 Rest point, 642 Rest solution, 177 Restocking a fishery, 86 Reversing the order of integration, 537 Review of power series, 262–263 Riccati’s differential equation, 69 Riemann mapping theorem, 928 Riemann sum, 564 Right-hand rule, 340 RK4 method, 72, 304 RK4 method for systems, 309, 311 RKF45 method, 305 Robin condition, 714 Robins, Benjamin, 205, 207 Rocket motion, 28, 191 Rodrigues’ formula, 290 Roll, 375 Root mean square, 681 Root test, 880 Roots of a complex number, 825–826 Rope pulled upward by a constant force, 31 Rotational flow, 514 Rotating fluid, shape of, 29 Rotating pendulum, 668 Rotating rod with a sliding bead, 204 Rotating shaft, 176 Rotating string, 171–172 Rotation and translation, 913 Rotation in the z-plane, 913 Round-off error, 298 Row-echelon form, 381 Row equivalent matrices, 381 Row operations, use in finding the inverse of a nonsingular matrix, 409 Row reduction, 381 Row space, 389 Row vector, 368, 389 Row vector form of an autonomous system, 631 R3, 329 R2, 323 Rules of differentiation, 833, APP-2 Runge–Kutta methods: first-order, 302 fourth-order, 72, 302 second-order, 303 Runge–Kutta–Fehlberg method, 305 Rutherford, Ernest, 76 INDEX Quadratic form: definition of, 450 as a matrix product, 450 Qualitative analysis: of first-order differential equations, 34, 36 of second-order differential equations, 150, 643–644, 652–654 of systems of differential equations, 600–601, 629 Quasi frequency, 158 Quasi period, 158 Quotient Rule, 833, APP-2 Reactance, 163 Reactions, chemical, 21–22, 87 Real axis, 822 Real integrals, evaluation by residues, 902–907 Real part of a complex number, 820 Real power function, 914 Real vector space, 353 Real-valued function, periodic, 676 Reciprocal lattice, 344 Rectangular coordinates, 327–328 Rectangular pulse, 235 Rectified sine wave, 247 Rectifying plane, 492 Recurrence relation: three term, 268 two term, 266 Reduced row-echelon form of a matrix, 381 Reduction of order, 117–119 Reflecting surface, 28–29 Region: closed, 829 in the complex plane, 829 connected, 527 disconnected, 527 with holes, 549 image of, 581 of integration, 534 invariant, 662 multiply connected, 527 open, 527 simply connected, 527 type I (II), 535, 662 Regression line, 92 Regular singular point of an ordinary differential equation: definition of, 272 solution about, 273 Regular Sturm–Liouville problem: definition of, 693 properties of, 693 Relative error: definition of, 72 percentage, 72 Relative growth rate, 84 Removable singularity, 894, 895 Repeller, 39, 601, 642 Residue(s): definition of, 897 evaluation of integrals by, 900, 902 at a pole of order n, 898 at a simple pole, 898 Residue theorem, 900 Resistance, 23 Resonance, pure, 160–161 Resonance curve, 166 Resonance frequency, 166 Response: definition of, 57 impulse, 250 of a series circuit, 78 of a system, 25, 115, 631 S Saddle point, 638 Sample point, 516, 534, 554, 564 Sampling Theorem, 793 Sawing wood, 91 Sawtooth function, 247 Scalar, 322, 672 Scalar acceleration, 488 Scalar matrix, 373 Index I-15 INDEX I-16 Scalar multiple: of a matrix, 369 of vectors, 322, 323, 329 Scalar triple product, 342 Scaling, 440 Schwartz, Laurent, 250 Schwarz–Christoffel transformations, 928–929 Second derivative of a complex function, 870–871 Second moments, 539 Second-order boundary-value problem, 169, 313, 316, 693 Second-order chemical reaction, 22, 87–88 Second-order DE as a system, 270, 309 Second-order difference equation, 429 Second-order differential operator, 178 Second-order initial-value problem, 14, 106, 309 Second-order Runge–Kutta method, 303 Second shifting theorem, 230 Second translation theorem: alternative form of, 231 form of, 230 inverse form of, 230 Self-adjoint form of a linear secondorder DE, 696 Semi-stable critical point, 39 Separable first-order differential equation: definition of, 43 solution of, 44 Separable partial differential equations, 708 Separated boundary conditions, 693 Separation constant, 709 Sequence: convergent, 878 criterion for convergence, 878 definition of, 878 Sequence of partial sums, 262 Series (infinite): absolutely convergent, 262, 880 circle of convergence, 881 complex Fourier, 689 of complex numbers, 878 convergent, 262, 879 cosine, 683 definition of, 878 Fourier, 678 Fourier–Bessel, 700 Fourier–Legendre, 702 geometric, 879 interval of convergence, 262 Laurent, 888–889 Maclaurin, 263, 884, 885 necessary condition for convergence, 879 nth term test for, 880 orthogonal, 674–675 power, 262, 881 Index radius of convergence, 262, 881 solutions of ordinary differential equations, 261 sine, 683 Taylor, 263, 882–884 tests for convergence, 262, 880 trigonometric, 677 Series circuits, 23, 78, 161–162 Sets in the complex plane, 828–829 Shaft through the Earth, 28 Shifting summation index, 263 Shooting method, 316 Shroud of Turin, 75, 80 Sifting property, 250 Signal processing, 793 Similar matrices, 416 Simple closed curve, 516 Simple harmonic electrical vibrations, 162 Simple harmonic motion, 152 Simple pendulum, 189 Simple pole, 894 Simply connected domain, 527, 859 Simply connected region, 527 Simply supported end of a beam, 168 Simply supported end conditions of a beam, 168 Sine integral function, 58, 768, 782 Sine series, 683 Sine series in two variables, 743 Single-step numerical method, 307 Singular boundary-value problem, 695 Singular matrix, 406, 409 Singular point of a complex function: definition of, 887 essential, 895 isolated, 887 nonisolated, 888 pole, 894 removable, 894 Singular point of a linear ordinary differential equation: definition of, 53, 264, 265 at infinity, 265 irregular, 272 regular, 272 solution about, 271, 273 Singular solution, 10 Singular Sturm–Liouville problem: definition of, 693 properties of, 693 Sink, 514 Sinking in water, 90 SIR model, 99 Skew-symmetric matrix, 405 Skydiving, 27, 82, 92 Sliding bead, 653 Sliding box on an inclined plane, 83 Slope field, 34 Smooth curve, 516 Smooth function, 482 Smooth surface, 552 Snowplow problem, 29 Soft spring, 188 Solar collector, 90 Solenoidal vector field, 514 Solution curve: of an autonomous differential equation, 37 definition of, Solution of a linear second-order partial differential equation: definition of, 708 particular, 708–709 Solution of a linear system of DEs, methods of, 197, 593 Solution of an ordinary differential equation: about an ordinary point, 265 about a regular singular point, 273 definition of, domain of, existence and uniqueness of, 15–16, 112 explicit, general, 11, 53, 112, 113 implicit, integral-defined, 10–11, 55, 58 interval of definition, losing a, 45 nontrivial, 169 n-parameter family of, particular, 9, piecewise-defined, 10 singular, 10 trivial, verification of a, 7, Solution of a system of differential equations, 10, 197, 593 Solution of a system of linear algebraic equations: definition of, 377, 384 number of parameters in a, 391–392 Solution space, 356 Solution vector, 593 Source, 514 Space curve: definition of, 480 length of, 484 Span, 357 Spanning set, 357 Sparse matrix, 804 Specific growth rate, 84 Speed, 486 Spherical Bessel function: of the first kind, 287 of the second kind, 287 Spherical coordinates: conversion to cylindrical coordinates, 570–571 conversion to rectangular coordinates, 570–571 definition of, 570 Laplacian in, 760 triple integrals in, 571 Spherical wedge, 571 Subscript notation, Subspace: criteria for, 354 definition of, 354 Substitutions: in differential equations, 65 in integrals, 544, 580 Subtraction of vectors, 323, 329 Successive mappings, 914 Sum of square errors, 469 Sum Rule, 833, APP-2 Summation index, shifting of, 263 Superposition principle: for BVPs involving the wave equation, 722 for Dirichlet’s problem for a rectangular plate, 727–728 for homogeneous linear ODEs, 109 for homogeneous linear PDEs, 709 for homogeneous systems of linear algebraic equations, 386 for homogeneous systems of linear DEs, 594 for nonhomogeneous linear ODEs, 114 Surface, orientable, 555–556 Surface area: differential of, 554 as a double integral, 553 Surface integral: applications of, 554, 556 definition of, 554 evaluation of, 554 over a piecewise defined surface, 557 Suspended cable, mathematical model of, 24–25, 49, 190–191 Suspension bridge, 24, 49, 190 Sylvester, James Joseph, 351 Symmetric equations for a line, 346 Symmetric matrix: definition of, 373 eigenvalues for, 430 eigenvectors of, 604 orthogonality of eigenvectors, 431–432 Syndrome, 465 Systematic elimination, 197 Systems of DEs: higher-order DEs reduced to, 309, 630 numerical solution of, 309, 311 reduced to first-order systems, 310 Systems of linear algebraic equations: as an augmented matrix, 380 coefficients of, 377 consistent, 377 elementary operations on a, 380 homogeneous, 377, 384 ill-conditioned, 417 inconsistent, 377 Gaussian elimination, 381 Gauss–Jordan elimination, 381 general form of, 377 matrix form of, 385 nonhomogeneous, 377 overdetermined, 386 solution of, 377 superposition principle for, 386 underdetermined, 386 Systems of first-order differential equations: autonomous, 630, 631 definition of, 10, 93, 592, 630 linear form of, 592 matrix form of, 592, 631 solution of, 10, 593 Systems of linear algebraic equations, methods for solving: using augmented matrices, 380–381 using Cramer’s rule, 415–416 using elementary operations, 378 using elementary row operations, 380–381 using the inverse of a matrix, 411 using LU-factorization, 456 Systems of linear first-order differential equations, methods for solving: using diagonalization, 611, 619 using the Laplace transform, 251, 623 using matrices, 598–608 using a matrix exponential, 621–623 using systematic elimination, 197–198 using undetermined coefficients, 614 using variation of parameters, 616–617 T Tables: of conformal mappings, APP-9 of derivatives and integrals, APP-2 of Laplace transforms, 215, APP-6 of trial particular solutions, 131 Tangent line, 70 Tangent plane to a surface: definition of, 508 equation of, 508 vector equation of, 508 Tangent vectors, 482, 491–493 Tangential component of acceleration, 492 Taylor series, 149, 883 Taylor’s theorem, 884 Telegraph equation, 716 Telephone wires, shape of, 24–25, 190–191 Temperature: in an annular cooling fin, 291–292 in an annular plate, 176, 751 in a circular plate, 748, 764 in a circular cylinder, 756–757, 764 between concentric cylinders, 146 between concentric spheres, 175, 762 in a cooling/warming body, 21, 76, 80–81 in an infinite cylinder, 758 in an infinite plate, 763 in a one-eighth annular plate, 752 in a quarter-annular plate, 752 in a quarter-circular plate, 751 Index INDEX Spiral points: stable, 640 unstable, 640 Spread of a disease, 21 Spring constant, 152 Spring coupled pendulums, 259 Spring/mass systems, 152–161 Spring pendulum, 205–206 Square errors, sum of, 469 Square matrix, 368 Square norm of a function, 673 Square wave, 247 Stability criteria: for first-order autonomous equations, 646 for linear systems, 642 for plane autonomous systems, 647 Stability for explicit finite difference method, 809, 814 Stability of linear systems, 636 Stable node, 638, 642 Stable numerical method, 308 Stable critical point, 644 Stable spiral point, 640, 642 Staircase function, 235 Standard basis: for Pn, 355 for R2, 325, 355 for R3, 330, 355 for Rn, 356 Standard Euclidean inner product, 352 Standard form for a linear differential equation, 51, 118, 137, 264, 272 Standard inner product in Rn, 352 Standing waves, 721, 755 Starting methods, 307 State of a system, 20, 25, 472, 631 State variables, 25 Stationary point, 36 Steady-state current, 79, 162 Steady-state fluid flow, 938 Steady-state solution, 79, 159, 162, 732 Steady-state temperature, 713, 725, 748, 750, 756, 760 Steady-state term, 79, 159 Stefan’s law of radiation, 101 Step size, 71 Stochastic matrix, 426 Stokes, George G., 559 Stokes’ law of air resistance, 207 Stokes’ theorem, 559 Stream function, 938 Streamlines, 65, 861 Streamlining, 939 String falling under its own weight, 771–772 String of length n, 463 Sturm–Liouville problem: definition of, 693 orthogonality of solutions, 693 properties of, 693 regular, 693 singular, 695 Submatrix, 376 I-17 INDEX I-18 Temperature:—(Cont.) in a rectangular parallelepiped, 744 in a rectangular plate, 725 in a rod, 716 in a semiannular plate, 752 in a semicircular plate, 750 in a semi-infinite plate, 729 in a sphere, 760, 762 in a wedge-shaped plate, 751 Terminal velocity of a falling body, 43, 82, 90 Test point, 924 Theory of distributions, 250 Thermal conductivity, 712 Thermal diffusivity, 713 Three-dimensional Laplacian, 712 Three-dimensional vector field, 495–496 3-space (R3), 329 Three-term recurrence relation, 268 Threshold level, 89 Time of death, 81 TNB-frame, 493 Torque, 343 Torricelli, Evangelista, 206 Torricelli’s law, 23 Trace: of a matrix, 636 of a plane, 348 Tracer, 472 Tractrix, 28, 101 Trajectories, orthogonal, 102 Trajectory, 593, 600, 631 Transfer coefficients, 473 Transfer function, 224 Transfer matrix, 473 Transform pair, 783 Transformation, 581 Transient solution, 159, 162 Transient term, 79, 159 Translation: and rotation, 913 in the z-plane, 913 Translation on the s-axis, 226 Translation on the t-axis, 230 Translation property for autonomous DEs, 40 Translation theorems for Laplace transform, 226, 230 Transpose of a matrix: definition of, 371 properties of, 372 Transverse vibrations, 713 Traveling waves, 724 Tree diagrams, 499 Triangle inequality, 822 Triangular matrix, 372 Triangular wave, 247 Tridiagonal matrix, 810 Trigonometric functions, complex: definitions of, 846 derivatives, 846 Trigonometric identities, 846 Trigonometric series, 677 Index Triple integral: applications of, 566 in cylindrical coordinates, 569 definition of, 564–565 evaluation of, 565–566 in spherical coordinates, 571 as volume, 566 Triple scalar product, 342 Triple vector product, 342 Triply connected domain, 859 Trivial solution: defined, for a homogeneous system of linear equations, 412 Trivial vector space, 353 Truncation error: for Euler’s method, 299 global, 300 for improved Euler’s method, 301 local, 299 for RK4 method, 305 Tsunami, mathematical model of, 90 Twisted cubic, 494 Twisted shaft, 739 Two-dimensional definite integral, 534 Two-dimensional fluid flow, 511, 514, 831 Two-dimensional heat equation, 741 Two-dimensional Laplace’s equation, 712 Two-dimensional Laplacian, 712 Two-dimensional vector field, 510–511 Two-dimensional wave equation, 712 Two-point boundary-value problem, 107, 169 2-space (R2), 323 Two-term recurrence relation, 266 Type I (II) invariant region, 662 Type I (II) region, 535 U Uncoupled linear system, 611 Undamped forced motion, 160 Undamped spring/mass system, 152, 160 Underdamped electrical circuit, 162 Underdamped spring/mass system, 156 Underdamped system, 654 Underdetermined system of linear algebraic equations, 386 Undetermined coefficients: for linear differential equations, 127–134 for linear systems, 614 Uniqueness theorems, 16, 106 Unit impulse, 248 Unit step function: definition of, 229 graph of, 229 Laplace transform of, 230 Unit tangent, 491 Unit vector, 324 Unstable critical point, 39, 644, 646 Unstable node, 638, 642 Unstable numerical method, 308, 809 Unstable spiral point, 640, 642 Unsymmetrical vibrations, 188 Upper triangular matrix, 372 Upward orientation of a surface, 556 USS Missouri, 176 V Van der Pol’s differential equation, 663, 664 Van der Waal’s equation, 501 Variable mass, 27, 191–192 Variable spring constant, 155 Variables, separable, 43, 66, 68 Variation of parameters: for linear DEs, 51, 136–140 for systems of linear DEs, 616–617 Vector(s): acceleration, 486 addition of, 322–323, 329 angle between, 334 binormal, 492 component on another vector, 335 components of a, 323, 329 in a coordinate plane, 323 coplanar, 343 cross product of, 338–339 difference of, 322, 323, 329 differential operator, 501 direction, 345 direction angles of, 334 direction cosines of, 334 dot product of, 332, 333 equality of, 322, 323, 329 equation for a line, 345 equation for a plane, 347 fields, 511 free, 322 function, 480 geometric, 322 horizontal component of, 325 initial point of, 311 inner product, 332, 352 length of, 324, 329, 352 linear combination of, 324 linearly dependent, 355 linearly independent, 355 magnitude of, 324, 329, 333, 352 multiplication by scalars, 322, 323, 324, 329, 352 negative of, 322 norm of, 324, 352 normal to a plane, 347–348 normalization of, 324, 352 in n-space, 352 orthogonal, 333 orthogonal projection onto a subspace, 361 orthonormal basis, 359 parallel, 322, 341 position, 323, 329 principal normal, 492 projection on another vector, 335 rules of differentiation, 483 smooth, 482 of three variables, 501, 510 of two variables, 501, 510 as velocity, 486 Vector-valued functions, 480 Vector space: axioms for a, 352 basis for a, 357 closure axioms for a, 353 complex, 353 dimension of, 356 finite dimensional, 356 infinite dimensional, 356 inner product, 357 linear dependence in a, 355, 357 linear independence in a, 355, 357 real, 353 span of vectors in a, 357 subspace of, 354 trivial, 353 zero, 353 Velocity field, 490 Velocity potential, complex, 938 Velocity vector function, 486 Verhulst, P F., 85 Vertical component of a vector, 325 Vibrating cantilever beam, 741 Vibrating string, 713 Vibrations, spring/mass systems, 152, 196–197 Virga, 29 Viscous damping, 24 Voltage drops, 23 Volterra integral equation, 241 Volterra’s principle, 658 Volume of a parallelepiped, 343 Volume under a surface: using double integrals, 535 using triple integrals, 566 Von Mises, Richard, 438 Vortex, 942 Vortex point, 642 W Water clock, 102 Wave equation: derivation of the one-dimensional equation, 713 difference equation replacement for, 812 one-dimensional, 711–712, 812 solution of, 719–720 two-dimensional, 742 Weight function: of a linear system, 250 orthogonality with respect to, 675 Weighted average, 302 Wire hanging under its own weight, 24, 190–191 Word: definition of, 463 encoding, 463 Work: as a dot product, 336 as a line integral, 521 Work done by a constant force, 336 Wronskian: for a set of functions, 111 for a set of solutions of a homogeneous linear DE, 111, 137 for a set of solutions of a homogeneous linear system, 595 Wronskian determinant, 111 X x-coordinate of a point in 3-space, 328 xy-plane, 328 xz-plane, 328 Y Yaw, 375–376 y-coordinate of a point in 3-space, 328 Young’s modulus, 168 yz-plane, 328 INDEX properties of, 324, 332, 339 right-hand rule, 340 scalar multiple of, 322, 323, 329 scalar product, 332 scalar triple product, 342 as a solution of systems of linear DEs, 593 span of, 357 spanning set for, 357 standard basis for, 325, 330, 356 subtraction of, 323, 329 sum of, 322 tangent to a curve, 486 terminal point of, 311 triple product, 342 in 3-space, 327 in 2-space, 322 unit, 324, 352 unit tangent to a curve, 491 vector triple product, 342 velocity, 486 vertical component of, 325 zero, 322, 329, 352 Vector differential operator, 501 Vector equation for a curve, 480 Vector equation for a line, 345 Vector equation of a plane, 348 Vector fields: and analyticity, 936 conservative, 525 curl of, 512 definition of, 510–511 divergence of, 512 flux of, 512 gradient, 511 irrotational, 514 plane autonomous system of, 631 rotational, 514 solenoidal, 514 three-dimensional, 510–511 two-dimensional, 510–511 velocity, 511 Vector functions: as acceleration, 486 continuity of, 481 definition of, 480 derivative of, 482 differentiation of components, 482 higher derivatives of, 483 integrals of, 484 limit of, 481 Z z-axis in space, 327–328 z-coordinate of a point in 3-space, 328 Zero matrix, 372 Zero vector, 322, 352 Zero vector space, 353 Zero-input response, 224 Zeros: of Bessel functions, 285 of complex cosine and sine, 846–847 of complex hyperbolic cosine and sine, 848 of a function, 896 of order n, 896 Zero-state response, 224 z-plane, 822 Index I-19 ... content of a course entitled engineering mathematics often varies considerably between two different academic institutions Therefore a text entitled Advanced Engineering Mathematics is a compendium... 978-1-284-10590-2 Library of Congress Cataloging-in-Publication Data Author: Zill, Dennis G Title: Advanced Engineering Mathematics / Dennis G Zill, Loyola Marymount University Description: Sixth edition |... endorsement purposes All trademarks displayed are the trademarks of the parties noted herein Advanced Engineering Mathematics, Sixth Edition is an independent publication and has not been authorized,
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