Dennis C. Zill - llrc.mcast.edu.mtllrc.mcast.edu.mt/digitalversion/Table_of_Contents_28104.pdf ·...

Second Edition

Dennis C. Zill Loyola Marymount University

Michael R. Cullen

JONES AND BARTLETT PUBLISHERS Sudbury, Massachusetts

BOSTON TORONTO LONDON SINGAPORE

a

Contents Contents ix

s'

3.6 Cauchy-Euler Equation 136

3.7 Nonlinear Equations 142

3.8 Linear Models: Initial-Value Problems 147 3.8.1 SpringMass Systems: Free Undamped Motion 147

Vectors. Matrices. and Vector Calculus 3.8.2 SpringMass Systems: Free Damped Motion 151

3.8.3 SpringMass Systems: Driven Motion 154

3.8.4 Series Circuit Analogue 157 7.1 Vectors in 2-Space 298 3.9 Linear Models: Boundary-Value Problems 163 7.2 Vectors in 3-Space 304 3.1 0 Nonlinear Models 170 7.3 The Dot Product 309

3.1 1 Solving Systems of Linear Equations 180 7.4 The Cross Product 3 17

Chapter 3 Review Exercises 186 7.5 Lines and Planes in 3-Space 323

4 The Laplace Transform 189 7.6 Vector Spaces 331

Chapter 7 Review Exercises 338 4.1 Definition of the Laplace Transform 190

4.2 The lnverse Transform and Transforms of Derivatives 195

4.3 Translation Theorems 204 8.1 Matrix Algebra 342 4.3.1 Translation on the s-axis 204

8.2 Systems of Linear Algebraic Equations 351 4.3.2 Translation on the t-axis 207

8.3 Rank of a Matrix 363 4.4 Additional Operational Properties 21 5

8.4 Determinants 368 4.5 Dirac Delta Function 224

8.5 Properties of Determinants 374 4.6 Solving Systems of Linear Equations 227

8.6 - lnverse of a Matrix 381 Chapter 4 Review Exercises 232 8.6.1 Finding the Inverse

5 Series Solutions of Linear Equations 235 8.6.2 Using the lnverse to Solve Systems 387

8.7 Cramer's Rule 391 5.1 Solutions about Ordinary Points 236 8.8 The Eigenvalue Problem 395

5.1.1 Review of Power Series 236 8.9 Powers of Matrices 400

5.1.2 Power Series Solutions 238 8.1 0 Orthogonal Matrices 404

5.2 Solutions about Singular Points 246 i

8.1 1 Approximation of Eigenvalues 41 1 5.3 Two Special Equations 256

8.1 2 Diagonalization 41 8 Chapter 5 Review Exercises 290 8.1 3 Cryptography 427

6 Numerical solutions of Ordinary Differential 8.1 4 An ~rror-correcting Code 430

Equations 267 8.1 5 Method of Least Squares 436

6.1 Euier Methods and Error Analysis 268 8.1 6 Discrete Compartmental Models 439

6.2 Runge-Kutta Methods 273 Chapter 8 Review Exercises 444

6.3 Methods 278 9 Vector Calculus 447 6.4 Higher-Order Equations and Systems 281

9.1 Vector Functions 448 6.5 second-order Boundary-Value Problems 286

9.2 Motion on a Curve 454 Chapter 6 Review Exercises 290

9.3 Curvature and Components of Acceleration 459

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Contents

C'

9.4 Functions of Several Variables 464

9.5 The Directional Derivative 470

9.6 Planes and Normal Lines 477

9.7 Divergence and Curl 480

9.8 Line lntegrals 486

9.9 Line lntegrals Independent of the Path 495

9.1 0 Review of Double lntegrals 502

9.1 1 Double lntegrals in Polar Coordinates 51 1

9.1 2 Green's Theorem 51 6

9.13 Surface Integrals 521

9.1 4 Stokes' Theorem 529

9.1 5, Review of Triple lntegrals 534

9.1 6 Divergence Theorem 546

9.17 Change of Variables in Multiple lntegrals 552

Chapter 9 Review Exercises 558

Part 3 Svstems of Differential Equations 563

10 Systems of Linear Differential Equations 567 10.1 Preliminary Theory 568

10.2 Homogeneous Linear Systems 575 10.2.1 Distinct Real Eigenvalues 576

10.2.2 Repeated Eigenvalues 579

10.2.3 Complex Eigenvalues 583

10.3 Solution by Diagonalization 588

10.4 Nonhomogeneous Linear Systems 590

10.5 Matrix Exponential 595


I I 11 Systems of Nonlinear Differential Equations 601

1 1.1 Autonomous Systems, Critical Points, and Periodic Solutions 602

1 1.2 Stability of Linear Systems 608

1 1.3 Linearization and Local Stability 61 7

1 1.4 Modeling Using Autonomous Systems 626

1 1.5 Periodic ~ o l u t i o n s f ~ i m i t Cycles, and Global Stability 635


Contents

Part 4 .." .T7.,,.,-, - ,.., . - .,,-,..,...,,.... , . - ,:;;i,',.*..'.**,;: , ,,,'., ; 2 . 2 . 2'. . . . ' . .

Fourier Series and Partial Differential Equations 647

12 Orthogonal Functions and Fourier Series 651

12.1 Orthogonal Functions 652

12.2 Fourier Series 656

12.3 Fourier Cosine and Sine Series 662

12.4 Complex Fourier Series and Frequency Spectrum 669

12.5 Sturm-Liouville Problem 672

12.6 Bessel and Legendre Series 678 12.6.1 Fourier-Bessel Series 678

12.6.2 Fourier-Legendre Series 682


13 Boundary-Value Problems in Rectangular Coordinates 687

13.1 Separable Partial Differential Equations 688

1 3.2 Classical Equations and Boundary-Value Problems 692

13.3 Heat Equation 697

13.4 Wave Equation 700

13.5 Laplace's Equation 704

13.6 Nonhomogeneous Equations and Boundary Conditions 709

13.7 Orthogonal Series Expansions 71 1

13.8 Fourier Series in Two Variables 71 5

Chapter 13 Review Exercises 71 7

14 Boundary-Value Problems in Other Coordinate Systems 719

14.1 Problems Involving Laplace's Equation in Polar Coordinates 720

14.2 Problems in Polar and Cylindrical Coordinates: Bessel Functions 724

14.3 Problems in Spherical Coordinates: Legendre Polynomials 730


Contents

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15 lntegral Transform Method 735

15.1 Error Function 736

15.2 Applications of the Laplace Transform 737

15.3 Fourier lntegral 745

15.4 Fourier Transforms 750

15.5 Fast Fourier Transform 756


76 Numerical Solutions of Partial Differential Equations 767

16.1 Elliptic Equations 768

16.2 Parabolic Equations 773

16.3 Hyperbolic Equations 779


Part 5 *-.. ".. .., . .,, , . .,. ,. - , , , .. . . , '<, ; ; , ' . .2.*;-..;..:- ... ; . .... . .. .

17 Functions of a Complex Variable 789

17.1 Complex Numbers 790

17.2 Form of Complex Numbers; Powers and Roots 793

17.3 Sets of Points i n the Complex Plane 798

17.4 Functions of a Complex Variable; Analyticity 800

17.5 Cauchy-Riemann Equations 806

17.6 Exponential and Logarithmic Functions 81 1

17.7 Trigonometric and Hyperbolic Functions 81 8

17.8 Inverse Trigonometric and Hyperbolic Functions 822


18 Integration in the Complex Plane 827 18.1 Contour Integrals 828

18.2 Cauchy-Goursat Theorem 833

18.3 Independence of Path 838

18.4 Cauchy's lntegral Formula 845


Contents xiii

,i , "4 '.I 1 19 Series and Residues 851 19.1 Sequences and Series 852

19.2 Taylor Series 857

19.3 Laurent Series 863

19.4 Zeros and Poles 870

19.5 Residues and Residue Theorem 873

19.6 Evaluation of Real Integrals 879


20 Conformal Mappings and Applications 889 20.1 Complex Functions as Mappings 890

20.2 Conformal Mapping and the Dirichlet Problem 894

20.3 Linear Fractional Transformations 902

20.4 Schwarz-Christoffel Transformations 908

20.5 Poisson lntegral Formulas 91 3

20.6 Applications 9 1 7


Appendix I Some Derivative and Integral Formulas A-3

Appendix I1 Gamma Function A-5

Appendix I 1 Exercises A-6

Appendix Ill Table of Laplace Transforms A-7

Appendix IV Conformal Mappings A-1 1

Appendix V Some BASIC Programs for Numefical Methods A-19

Selected Answers for Odd-Numbered Problems A-25

a I Index

a' Absolute convergence:

reflecting Property of a parabolic mirror, 32 of second kind, 258 Characteristic functions, 166

of a complex series, 854 restocking, 85 spherical, 261 Characteristic values, 395 of a power series, 236 rocket motion, 178 Bessel's differential equation, 256 Characteristic Vectors, 395

Absolute errot, 71 rotating string, 167-168 Chebyshev's differential equation, 685

parametric, 259 slipping chain, 27 solution of, 256-257

Chemical equations, balancing of, 360

snowplow problem, 32 ~ i h m o n i c function, 902 Chemical reactions: spread of a disease, 23 Binary strings, 430

first order, 23

s~rindmass systems, 30, 147-156, 170-171 Binormal, 461 second order, 24, 86 suspended wire, 173 Bits, 430 Circle:

tapered column buckling under a compressive force, 255 Boundary: in complex plane, 798 Adjoint matrix, 382

temperature distributions, 692-694 conditions, 103 of convergence, 855 use in find an inverse, 383

terminal velocity, 43, 81, 89 point, 768,799 of curvature, 462 Aging spring, 150 tractrix, 3 1 of a set, 800 Circulation, 493, 83 1 Agnew, Ralph Palmer, 32,121 tsunami, 90 Classification of a differential equation: Air resistance, 26, 29, 81, 89 ~o~ndary-value problem(s), 103

variable mass, 30, 175 linear models of, 163 as linear or nonlinear, 7 Ajr ' s differential equation, 150,242,259, vibrating string, 693-694, 700-702 solution of, 286-287,289 I i Algebraic equations, 35 1 Ali, Sherif F., 650

~ r c , 605 Bounding theorem for complex integrals, 830 by type, 6 Closed curve, 486 Archimedes' principle, 30 I

Amplitude of vibrations, 148-149 BOX product, 320 Area: Closed form solutions, 13

Branch: Analyuc function, 805 Closure axioms of a vector space, 332 as a double integral, 503 of the complex logarithm, 8 16 criterion for, 808 I as a double integral in polar coordinates, 514 cut, 816 Code, 43 1 I

derivatives of, 845-846 of a parallelogram, 320 Breidenthd, Robert E., 788 Code word, 43 1

Analyticity at a point, 805 Cofactor, 369 of a surface, 522 Buckling of a thin vertical col~mn, 166,169 Angle between two vectors, 3 11 Cofactor expansion of a determinant, 372 I

Angle preserving mappings, 894 Argument of a complex number, 794

column vector, 343,363 ! principal, 794

I I

Anharmonic overtones, 730 Arithmetic modulo 2, 430 Cantilever beam, 164 Commutator of two matrices, 444 i Annulus in the complex plane, 799

Asymptotically stable critical point, 41, 617 Carbon dating, 74-75 compartmental analysis, 439 Antiderivative, 6 Competition models, 94,63 1 1 I I Attractor, 42, 578

of a complex function, 839 Carrying capacity of environment, 83 Augmented matrix, 354-355 Complementary error function, 56,736 I

existence of, 841 Cartesian coordinates, 304-305 reduced row-echelon form of, 356 Catenary, 174

Complementary function, 110 I Applications of differential equations: row-echelon form of, 355-356 1 caUchy-Euler differential eq~ation, 136 Complete set of functions, 655-656

aging springs, 150,265 Autonomous differential equations, 38, 146 method of solution, 137-139 Complex form of Fourier series, 670

air resistance, 26, 29, 89 A~t~nomous systems of differentid equations, 602 Cauchy-Goursat theorem, 834 Complex function:

atomic physics, 22 Auxiliary equations: analytic, 805

I biology, 22, 93-95

1 I

for linear constant-coefficient differential equations, 117, 119 continuous, 803 buckling of a thin vertical column, 166, 169

1 for Cauchy-Euler equations, 137 &fintion of, 800-801 . i

carbon dating, 74-75 Cauchy-Riemann equations, 807 Axis of symmetry, 163 derivative of, 804 i

chemistry, 23-24,85-86 Cauchy-Schwarz inequality, 316 Cauchy's inequality, 847 differentiable, 804 1

domain of, 800 i competing species of animals, 94-95 Cauchy's integral formula, 843 continuous compound interest, 22 Back substitution, 354 for derivatives, 845 entire, 805

deflection of beams, 163-165,210-211 exponential, 812 Backward difference, 287 I Cauchy's residue theorem, 876 I double pendulum, 229-230

Balancing a chemical equation, 360 ~ Cauchy's theorem, 833 hyperbolic, 820 draining of a tank, 24-25, 88-89 Banded matrix, 771 Cayley, Arthur, 331,342 inverse hyperbolic, 824 electric circuits, 25, 29, 77-78,

BASIC programs, 945 Cayley-Hamilton theorem, 400 inverse trigonometric, 822 evaporation, 90 limit of, 802 falling bodies, 26, 89,

Basis of a vector space, 335 Center, 613 standard, 335 of curvature, 463 logarithmic, 814

floating barrel, 30 Beams: of mass, 506507,537 as a mapping, 801 harvesting, 85, 88 polynomial, 803 I

static deflection of, 163-165 Central difference, 287 I hole drilled through earth, 3 1 vibrating, 703, 714 approximation for derivatives, 287 range of, 801 I

immigration, 88 Beats, 162 Central force, 458 rational, 803 1

logistic population growth, 83-84 Bendixson negative criterion, 636 Centripetal acceleration, 455 trigonometric, 8 18-8 19 memorization, 3 1

Bernoulli's differential equation, 66 as a two-dimensional fluid flow, 801 meteorology, 32 Centroid, 507,528, 537

Bessel functions: chain rule of partial derivatives, 467 Complex number(s1: mixtures, 24, 76, 92

differential recurrence relations for, 260-261 Change of variables: addition of, 790 I I

networks, 95,228-229 argument of, 794 I pacemaker, 99

of first kind, 257 in a definite integral, 552 graphs of, 258 in a double integral, 513,552

associative laws for, 790 1 predator-prey, 93 laws for, 790 I

numerical values of, 259-260 in a triple integral, 540,542,556 pursuit curves, 179 3 properties of, 259 Characteristic equation of a matrix, 396 complex powers of, 817

lndex

Complex number(s) (continued) conjugate of, 791 definition of, 790 distributive laws for, 790 division of, 790 equality of,790 geometric interpretation of, 792 imaginary part of, 790 modulus of, 792 multiplication of, 790 polar form of, 793 integer powers of, 795 principal argument of, 794 principal nth root of, 797 pure imaginary, 790 real part of, 790 roots of, 796 subtraction of, triangle inequality for, 792 vector interpretation for, 792

Complex impedance, 814 Complex integral, 828 Complex plane, 792

imaginary axes of, 792 real axis of, 792

Complex potential, 919 Complex powers, 817 _

principal value of, 817 Complex sequence, 852 Complex series, 853 Complex vector space, 332 Complex velocity potential, 920 Component of a vector on another, 312-313 Components of a vector, 299,306,331 Conformal mapping, 894-895

and the Dirichlet problem, 898 Conformal mappings, table of, 937 Conjugate:

harmonic functions, 810 of a complex number, 791

Connected set, 800 Conservative vector field, 499, 501, 919 Consistent system of linear equations, 352 Continuing method, 278 Continuity:

equation of, 550 of a function of a complex variable, 803

Continuous compound interest, 22 Contour, 828

indented, 884 Contour integral(s), 828

fundamental theorem for, 840 method of evaluation, 828-829 properties of, 830

Convergence: of a Fourier-Bessel series, 681 of a Fourier integral, 746 of a Fourier-Legendre series, 682-683 J

d of a Fourier series, 659 of an improper integral, 190 of a power series, 236

Convolution: integral, 216,754

theorem, 216,754 Cooling, Newton's Law of, 23,75 Coordinate planes, 305 Coplanar vectors, 321 - Coulomb's law, 528 Coupled springs, 180 Coupled systems, 180 Cover-up method, 202 Cramer's rule, 391-392 Crank-Nicholson method, 776 Critical damping, 151 Critical loads, 167 Critical points for autonomous first-order equations, 38

asymptotically stable, 41 atrractor, 42 repeller, 42 semistable, 42 unstable, 42

Critical points for autonomous linear systems: attractor, 578 center, 613 definition of, 604 degenerate nodes, 612-613 locally stable, 608 repeller, 578 saddle point, 61 1 stable node, 610-61 1, stable spiral point, 613 unstable, 608 unstable node, 61 1 unstable spiral point, 613

Critical points for plane autonomous systems: asymptotically stable, 617 definition of, 604 globally stable, 635 locally stable, 617 stable, 617 unstable, 618

Critical speeds, 169 Critically damped circuit, 157 Cross product, 3 17

properties of, 3 17 Cross ratio, 905 Cryptography, 427 Curl of a vector field, 482

physical interpretation of, 483-484 Curvature, 459 Curves:

closed, 486 peicewise smooth, 486 positive direction on, 486 simple closed, 486 smooth, 486

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lndex

Cycle, 605 Cylindrical coordinates, 538-539

Laplacian in, 727

D' Alembert's solution, 704 7

Damped amplitude, 153 Damped motion, 15 1 Damping constant, 15 1 Damping factor, 15 1 Decay, radioactive, 22, Decoding a message, 427,431 Definite integral, 486 Deflation, method of, 415 Deflection curve, 163 Deformation of contours, 835 Degenerate nodes, 612-613 Dell () operator, 470-471 DeMoivre's formula, 795 Density-dependent hypothesis, 83 Derivative (differentiation) of a complex function:

of complex exponential functions, 812 definition of, 804 of complex hyperbolic functions, 820 of complex inverse hyperbolic functions,824 of complex inverse trigonometric functions, 823 of the complex logarithm, 8 16 of complex trigonometric functions, 8 19 rules for, 804

Derivatives of real functions: formulas, table of, 929 notation for, 6

Determinant(s): cofactors of, 369 defition of, 368,369,372 evaluating by row reduction, expansion by cofactors, 369-370 minor, 369 properties of, 374-379

Diagonalizable matrix, 419 Diagonalizability, criterion for, 419-420 Diagonally dominant matrix, 772 Diagonal matrix, 347 Difference equation:

replacement for heat equation, 774 replacement for Laplace's equation, 768 replacement for a second-order ODE, 287 replacement for wave equation, 779-780

Difference quotients, 287,768,774,780 finite differences, 287

Differential: of a function of several variables, 59,495,496 operator, 105 recurrence relation for Bessel functions, 260-261 of surface area, 523

Differential equation, 6 Bernoulli, 66 exact, 59

explicit solution of, 9 homogeneous, 64 implicit solution of, 10 as a mathematical model, 21 linear, 7,5 1, nonlinear, 8 normal form of, 7 order of, 7 ordinary, 6, partial, 6, separable, 44 solution of, 8 substitutions in, 64 trivial solution of, 9

Diffusion equation, 469,696 Dimension of a vector space, 335 Dirac delta function, 225

Laplace transform of, 225 Dirichlet condition, 694 Dirichlet problem, 706,769, 897-899

superposition principle for, 706-707 Direction:

angles, 312 cosines, 312 field, 36 vector, 323

Directional derivative: for functions of two variables, 472 for function of three variables, 473-474

Discrete compartmental models, 439 Discrete Fourier transform pair, 757 Discretization error, 269 Distance formula, 305 Distributions, 226 Divergence of a vector field, 483

physical interpretation of, 493-484 Divergence theorem, 546 Domain:

in the complex plane, 833 of a function of two variables, 464 of a solution of a differential equation, 8

Dominant eigenvalue, 41 1 eigenvector, 41 1

Dot notation, 7 Dot properties product, of, 309 3 10

in terms of matrices, 405 as work, 314

Double eigenvalues, 722 Double integral:

definition of, 503 evaluation in polar coordinates, of, 504 5 11

properties of, 503-504 reversing the order of integration in, 506

Double sine series, 716 Doubly connected domain, 833 Driven motion, 154

Index a

C Driving function, 57 Escape velocity, 178 First standing wave, 702 Duffing's differential equation, 177 Essential singularity, 870 First translation theorem, 204 Dulac negative criterion, 636 Euler load, 167 inverse form of, 204-205 Dynamical system, 27, 603 Euler's constant, 259 Five-point approximation for Laplace's equation, 768

Euler's formula, 11 8 Flexural rigidity, 164 Euler's methods, 69, Flow:

Effective spring constant, 159 for systems, 281,285 of heat, 692-693 Effective weight, 458 Evaluation of real integrals by residues, 879 steady-state fluid, 920

Even function, 662 Flux through a surface, 526 properties of, 662 Folia of Descartes, 15

approximation of, 4 1 1 Exact differential, 59,495-496 Forced motion, 154 of a boundary-value problem, 166 tests for, 59, Forcing function, 112 of a matrix, 395,398 Exact differential equation, 59 Forward difference, 287 of muliplicity rn, 579 Existence of Fourier transforms, 751 Fourier-Bessel series, 680-681

Eigenvector of a matrix: Existence of Laplace transform, 193 Fourier coefficients, 658 definition of, 395 Existence and uniqueness of a solution, 18, 102 Fourier cosine transform:

Expansion of a function: definition of, 75 1 orthogonal, 406 in a complex Fourier series, 670 operational properties of, 752

Elastic curve, 163 in a cosine series, 663 Fourier integral, 746 Electrical networks, 95,328-329 in a Fourier series, 658 complex form, 749 Electrical vibrations: in a Fourier-Bessel series, 680-681 conditions for convergence, 746

in a Fourier-Legendre series, 682 cosine form, 748 in a sine series, 663 sine form, 748

overdamped, 157 in terms of orthogonal functions, 654-655 Fourier-Legendre series, 682 simple harmonic, 157 Explicit finite difference method, 774 underdamped, 157 Explicit solution, 9

Elementary functions, 12 Exponential function: conditions for convergence, 659 Elementary matrix, 362 definition of, 8 12 cosine, 663 Elementary row operations, 355 derivative of, 812 generalized, 655

notation for, 356 fundamental region for, 813 sine, 663 Elimination method(s): period of, 8 13 in two variables, 716

systems of algebraic equations, 355 properties of, 813 Fourier sine transform: systems of ordinary differential equations, 18 1 Exponential order, 193 definition of, 751

Elliptic helix, 449 Exponents of a singularity, 250 operational properties of, 752 Fourier transforms, 751 Elliptic partial differential equation, 690 External force, 112-154

Ellison, John H., 566 Extreme displacement, 148 existence of, 75 1 Embedded beam, 164 Fourth-order Runge-Kutta method, 71,273,283 Encoding a message, 431 Free end conditions, 164,703 Entire function, 805 Falling bodies, mathematical model of, 26 Free motion: Epidemics, 23, 84, 97 Family of solutions, 10-1 1, damped, 151 Equation of motion, 148 Fast Fourier transform, 756 undamped, 147 Equidimensional equation, 137 Fibonacci sequence, 403 Frequency, 148 Equilibrium position, 147 Fick's law, 99 Frequency response curve, 162 Equilibrium point, 38 Finite difference equation, 287 Frequency spectrum, 671 Equilibrium solution, 39,604 Finite differences, 287 Fresnel sine integral, 58 Error correcting code, 432 Finite difference methods, 286-287,768,774,780 Frobenious, method of, 248-249 Error detecting code, 432 explicit, 774 Frobenius' theorem, 249 Error function, 56,736 implicit, 776 Full-wave rectification of sine, 223 Errors in numerical methods: Finite dimensional vector spaces, 336 Function:

First buckling mode, 167 complex, 800-801 complementary error, 736 First harmonic, 702

First moments, 536 error, 736 First normal mode, 702 even, 662

local truncation, 269,272,276 First-order chemical reactions, 23 excitation, 112 percentage, 71 First-order differential equations: forcing, 112

applications of, 21-25, 26-27,73, 82 odd, 662 solutions of, 44,51,59, 64, 68 three of more variables, 465

round-off, 269 First-order Runge-Kutta method, 274 two variables, 464

Fundamental angular frequency, 67 1 Fundamental frequency, 702 Fundamental mode of vibration, 702 Fundamental matrix, 590-591 Fundamental period, 656,670 Fundamental region, 8 13 Fundamental set of solutions:

of a linear differential equation, 108 of a system of linear differential equations, 572

Fundamental theorem: of algebra, 848

for line integrals, 496

. Galileo, 26 Gamma function, 195, Gaussian elimination, 355 Gauss-Jordan elimination, 355 Gauss' law, 919 Gauss-Siedel iteration, 771

BASIC program for, 946 Gauss' theorem, 546 Generalized factorial function, 93 1 Generalized Fourier series, 655 General solution:

of a differential equation, 12 of a linear differential equation, 53, 108, 110 of a system of linear differential equations, 572-573

Generalized Fourier series, 655 Generalized functions, 226 Geometric series, 853 Geometric vectors, 298 Gibbs phenomenon, 664 Globally stable critical point, 635 Gompertz differential equation,. 85 Gradient:

definition of, 470-471 field, 499 geometric interpretation, 477

Great circles, 543 Green's identities, 551 Green's theorem, 5 16 Growth and decay, 73 Growth constant, 74

Half-life, 74 Half-range expansions, 665 Half-wave rectification of sine, 223 Hamilton, William R., 33 1 Hamming (7,4) code, 432 Hamming (8,4) code, 435 Hanging wire, 173-174 Hard spring, 171 Harmonic function, 809

transformation theorem for, 898

A-88 Index

Heat equation: derivation of one dimensional equation, 692-693 one dimensional, 696 solution of, 697 two dimensional, 715

Hermite's differential equation, 678 Homogeneous boundary conditions, 674 Homogeneous differential equation:

ordinary, 51,64, 104 partial, 688

Homogeneous function of degree a, 64 Homogeneous systems of algebraic equations, 359

nontrivial solutions of, 359,366 trivial solutions of, 359, 366

Homogeneous systems of linear differential equations, 568 complex eigenvalues, 583-585 distinct real eigenvalues, 576-578 repeated eigenvalues, 579-582

Hooke's law, 30,147,170 Horvath, Arpad, 296 Hyperbolic functions, 820-821 Hyperbolic partial differential equation, 690

Identity matrix, 348 i, j vectors, 301 i, j, k vectors, 307 IU-conditioned system, 394 Imaginary axis, 792 Impedance, 158,814 Implicit finite difference methods, 776 Implicit solution, 10 Improved Euler method, 271 Impulse response, 227 Incompressible fluid, 484 Inconsistent system of linear equations, 352 Indefinite integral, 839 Indented contours, 884 Independence of path, 496

test for, 498, 500 Indicial equation, 250-251 Indicial roots, 250,252-253 Inductance, 25 Infinite dimensional vector space, 336 Initial conditions, 16, 102 &,

Initial-value problem, 16, 102 for systems of linear differential equations, 570

Inner product: space, 338 of tw% functions, 652 of two vectors, 309,405

Input, 57, 112, 147 Insulated boundaries, 695 Integral curve, 10 Integral of a differential equation, 10 Integral equation, 219 Integral formulas, table of, 930

C'

Integral transform, 212,750 Integrating factor, 52, 62, 63 Integration by parts, 842 Integrodifferential equation, 219 Interest, compounded continuously, 22 Interior mesh points, 287 Interior point of a set in the complex plane, 798 Interpolating function, 278

, Interval: of convergence, 236 of definition of a solution, 8 of existence, 8, of existence and uniqueness, 18

Invariant region, 639 Inverse transform, 750

Fourier cosine, 75 1 Fourier sine, 751 Fourier, 75 1 Laplace, 195,

Inverse hyperbolic functions, 824 derivatives of, 824

Inverse integral transform, 750 Inverse of a matrix:

definition of, 38 1 finding, by the adjoint method, 383 finding, by elementary row operations, 385

Inverse power method, 417 Inverse trigonometric functions, 822

derivatives of, 823 Invertible matrix, 3 8 1 Irregular singular point, 247 Irrotational flow, 484 Isolated singularity, 860 Iterated integral, 504, 535-536

Jacobian, 553 Jacobian matrix, 620 Joukowski airfoil, 894 Joukowski transformation, 894

Kepler's first law of planetary moton, 458 Kernel of a transform, 750 Kirchhoff's point and loop rules, 358 Kirchhoff's second law, 25, 77, 157

Lagrange's identity, 323 Laguerre's differential equation, 678 Laplace transform:

behavior as s -+ -, 201 convolution theorem for, 216 definition of, 191 of derivatives, 198

lndex

of differential equations, 199 differentiation of, 215 of Dirac delta function, 225 existence of, 193 of an integral, 21 8 inverse of, 195-196 linearity of, 192 of a partial derivative, 737-736 of a periodic function, 220 tables of, 933 translation theorems for, 204,208 of systems of differential equations, 227 of unit step function, 209

Laplace's partial differential equation, 469, 692 maximal principle for, 706 solution of, 705

Laplacian: in cylindrical coordinates, 727 in polar coordinates, 720 in rectangular coordinates, 692 in spherical coordinates, 730

Latitude, 543 Laurent series, 863 Laurent's theorem, 864 Law of mass action, 86 Lazer, A. C., 2-4 Least squares:

line, 90,436-437 method of, 436

Legendre polynomials, 263 graphs of, 263 p

properties of, 263-264 recurrence relation for, 264 Rodrigues' formula for, 266

Legendre's differential equation, 256 solution of, 262-263

Level curves, 47,464 Level of resolution of a mathematical model, 21 Level surfaces, 465 Lewis, Gilbert N., 4 Libby, W., 74-75 Limit cycle, 639,640-641 Limit of a function of a complex variable, 802 Lineal element, 36 Linear combination, 301 Linear dependence:

of functions, 106 of solution vectors, 571-572 of vectors, 334 L

Linear donor-controlled hypothesis, 440 Linear equation(s), 35 1

systems of, 351-352 Linear fractional transformation, 902 Linear independence

of functions, 106 of solutions, 107 of solution vectors, 571-572 of vectors, 334

Linearity property: of inverse Laplace transform, 196 of Laplace transform, 190

Linearization, 68, of a nonlinear differential equation, 173,619 of a nonlinear system of differential equations, 619-621

Linear momentum, 458 Linear ordinary differential equations, 7

applications of, 21-27,73-78,82-87,147-158, 170-176 complementary function for, 110 first-order, 5 1 general solution of, 53, 108, 110 higher-order, 101 homogeneous, 51,104 nonhomogeneous, 51, 104 particular solution of, 51, 110 standard forms of, 51, 132, 135 superposition principles for, 105, 11 1

Linear second-order partial differential equations: classification of, 690 definition of, 688 homogeneous, 688 nonhomogeneous, 688 solution of, 688 superposition principle for, 690

Linear spring, 170 Linear system(s):

of algebraic equations, 35 1 of differential equations, 91,568

Linear transform, 192 Line integrals:

in the complex plane, 828 definition of, 487 evaluation of, 487,488,490 fundamental theorem for, 496 independent of path, 496 . in the plane, 487 in space, 490 as work, 491-492

Line segment, 323 Lines in 3-space:

parametric equations, 323 symmetric equations, 324 vector equation, 323

Liouville's theorem, 848 Lissajous curve, 185,232 Locally stable critical point, 608, 617 Logarithm of a complex number:

branch of, 816 branch cut for, 816 definition of, 814 derivative of, 816 principal value of, 8 15 properties of, 8 16

Logistic differential equation, 83 Logistic function, 83-84

graph of, 84 Longitude, 543

A-90 Index lndex

Lotka and Volterra, predator-prey model, 93 LR series circuit, differential equation of, 29,77,221 LRC series circuit:

differential equation of, 25, 157 integrodifferential equation of, 2 19

LU-factorization of a matrix, 351,362

Maclaurin series, 237, 858 Magnification in the z-plane, 892 Malthus, Thomas, 22 Malthusian model, 22 Mass, 506-507, 537

center of, 506507,537 action, law of, 86

Mathematical models, 21, Matrix (matrices):

addition of, 343 adjoint of, 382 associative law, 346 augmented, 354-355 banded, 771 characteristic equation of, 396 column vector, 343 definition of, 342 determinant of, 368-372 diagonal, 347 diagonalizable, 419 diagonally dominant, 772 difference of, 344 distributive law, 346 dominant eigenvalue of, 41 1 eigenvalues of, 395 eigenvectors of, 395 elementary, 362 entries of, 342 equality of, 343 exponential, 595 fundamental, 590-59 1 identity, 348 inverse of, 38 1 Jacobian, 620 lower triangular, 347 LU-factorization, 351, 362 main diagonal entries of, 343 multiplication, 345 nilpotent, 404

L - nonsingular, 38 1

order of, 342 orthogonal, 407 orthogonally diagonalizable, 422 partitioned, 35 1 powers of, 400,426 rank of, 363 reduced row-echelon form of, 356 row-echelon form of, 355-356 row equivalent, 355 row vector, 343 -

d scalar, 347 scalar multiples of, 344 singular, 3 8 1 size, 342 skew-symmetric, 380 sparse, 77 1 square, 342 stochastic, 400 subhraction of, 344 symmetric, 348 transfer, 440 transpose of, 346 triangular, 347 tridiagonal, 777 upper triangular, 347 vector, 363 zero, 347

Matrix form of a system of linear algebraic equations, 387 Matrix form of a system of linear differential equations, 569 Maximum principle, 706 Maxwell's equations, 485 McKenna, P.J., 2-4 Meander function, 223 Meridian, 543 Mesh:

points, 287,768 size, 768

Message, 427,430 Method of deflation, 415 Method of Frobenius, 248-249 Method of least squares, 436 Method of undetermined coefficients, 124 Midpoint of a line segment, 306 Minor, 369 Mixed partial derivatives, 466

equality of, 466,468 Mixtures, 24,76, 92 ML-inequality, 830-83 1 Mobius strip, 525 Modulus of a complex number, 792 Moments of inertia, 508,537

polar, 510 Multiply connected:

domain, 833 region, 497

Multistep numerical method, 278

n-dimensional vector, 33 1 Negative direction on a curve, 516 Neighborhood, 798 Net flux, 83 1 Neumann condition, 694 Newton's law of cooling/warming, 23,75 Newton's law of universal gravitation, 3 1 Newton's laws of motion:

first, 25 second, 26,30,31

Nodes: of a plane autonomous system, 610-61 1,612-613 of a standing wave, 702

Nonconservative force, 501 Nonelementary integral, 56 Nonhomogeneous linear differential equation

ordinary, 51, 104 partial, 688

Nonhomogeneous systems of algebraic equations, 352 Nonhomogeneous systems of linear differential equations, 568 Nonlinear ordinary differential equations, 8,82, 142,170 Nonlinear spring, 17 1 Nonsingular matrix, 38 1 Norm:

of a column vector (matrix), 405 of a function, 653 of a partition, 486, 487, 503, 523,535 of a vector, 300, 306, 331

Normal component of acceleration, 460-461 Normal form:

of an ordinary differential equation, 7 of a system of linear first-order equations, 568

Normalized set of functions, 654 Normal line to a surface, 479 Normal modes, 702 Normal plane, 461 Normal vector to a plane, 326 Normalized eigenvector, 415,423-424 Notation for derivatives, 6 n-parameter family of solGtions, 11 nth-order initial-value problem, 16, 102 nth-order system, 568 nth roots of unity, 765 nth term test for divergence, 854 Nullcline, 44 Numerical methods: Adams-BashforWAdams-Moulton, 278-279 adaptive, 277 errors in, 269-271,272, 276 Crank-Nicholson method, 776-777 deflation method, 415 Euler's method, 69 finite-difference methods, 286-287,768,780 Gauss-Seidel method, 77 1 improved Euler' s method, 27 1 inverse power method, 417 power method, 41 1-413 predictor-corrector, 27 1, Runge-Kutta methods, 7 1,273-274 shooting method, 289 stability of, 279, 776, 782

Numerical solver, 71

Octants, 305 Odd function, 662

properties of, 662

ODE, 6 One dimensional phase portrait, 39 One-to-one transformation, 553 Open disk, 798 Operator, differential, 105 Order of a differential equation, 7 Open set, 798 Ordinary differential equation, definition of, 6 Ordinary point of a differential equation, 239 Orientation:

of a curve, 486 of a surface, 525

Orthogonal curves, 100,810 Orthogonal eigenvectors, 406 Orthogonal functions, 652 Orthogonally diagonalizable matrix, 423 Orthogonal matrix, 404,407

constructing an, 408 Orthogonal series expansion, 654-655 Orthogonal set, 653

with respect to a weight function, 655 Orthogonal surfaces, 480 Orthogonal trajectories, 100 Orthogonal vectors, 3 11 Orthonormal set of functions, 653 Osculating plane, 461 Output, 57, 112, 147 Overdamped circuit, 157 Overdamped free motion, 15 1 Overdetermined system, 361,436 Overtones, 702

Parabolic partial differential equation, 690 Parallel vectors, 298, 318 Parametric Bessel equation, 259 Parity check bits, 431 Parity check equations, 433 '

Parity check matrix, 433 Partial derivatives:

Chain Rule for, 467 definition of, 465 higher-order, 466 mixed, 466 symbols for, 466

Partial differential equation, d e f i t i m of, 6 elliptic, 690,768 homogeneous, 688 hyperbolic, 690,779 linear, 688 nonhomogeneous, 688 parabolic, 690,773 separable, 688 solution of, 688

Particular solution, 11,51, 110 Partitioned matrix, 351 Path independence, 496

tests for, 496498,500 Pauli spin matrices, 444

Index

Pendulum: linear, 172-173

Period, 148,656 Periodic boundary conditions, 677 Periodic driving force, 666 Periodic extension of a function, 660 Periodic functions, Laplace transform of, 220 Periodic solution of a plane autonomous system, 605 Phase angle, 149

Phase plane, 577,609 Phase-plane method, 623 Phase portrait, 39,577,609 Piecewise continuous functions, 193 Piecewise smooth curve, 486

Planar transformation, 890 Plane autonomous system, 603

Cartesian equation, 325 normal vector to, 325, 326 vector equation, 325

Plucked string, 701,704 Poincare-Bendixson theorems, 640,642 Poisson integral formula:

for unit disk, 915 for upper half-plane, 914

Poisson's partial differential equation, 711,773 Polar form of a complex number, 793 Polar moment of inertia, 515

definition of, 870 of order n, 870, 872

Population dynamics, 22 Position vector, 299,306-307 Positive direction on a curve, 486,516

complex, 919 complex velocity, 920 energy, 502 function, 499

Power function, 892 Power method, 41 1-413 Power of a matrix, 400402 Power series:

center, 236, 855 circle of convergence, 855 definition of, 236,855 differentiation of, 237,858 integration of, 237, 857 - interval of convergence, 236

r ' Maclaurin, 858 radius of convergence, 236, 855 ratio test for, 236, 854-855, 856 represents an analytic function, 858 represents a continuous function, 237,857 review of, 236-237 solutions of differential equations, 238-239 Taylor, 858

Predator-prey, 93,629-63 1 Predictor-corrector methods, 27 1, 278 Prime meridian, 543 Prime notation, 6 Principal argument of a complex number, 794 Principal axes of a conic,425 Principal branch of the logarithm, 8 16 Principal normal, 460 Principal nth root of a complex number, 797 Principal part of a Laurent series, 870 Principal value:

of a complex power, 8 17 of an integral, 881 of logarithmic function, 815

Probability integral, 736 Projection of a vector onto another, 3 13-3 14 Pure resonance, 156 Pursuit curves, 179

Quadratic form, 424 Quasi frequency, 153 Quasi period, 153

R", 331 Radial vibrations, 724 Radial symmetry, 724 Radioactive decay, 22 Radioactive decay series, 92 Radius of convergence, 236,855 Radius of gyration, 508,537 Raleigh differential equation, 625 Rank of a matrix, 363

by row reduction, 364 Rational function, 803 Rational roots, 121 Ratio test, 236, 854-855 Rayleigh quotient, 413 RC series circuit, differential equation of, 77 Reactance, 158 Reactions, chemical, 23-24,85-87 Real axis, 792 Real vector space, 332 Reciprocal lattice, 3 16 Rectangular pulse, 213 Rectified sine wave, Rectifying plane, 461 Recurrence relation, 240,243 Reduced row-echelon form, 356

Reduction of order, 114, 143 Region:

closed, 800 connected, 800 in the complex plane, 800 invariant, 639 multiply connected, 497 open, 497 simply connected, 497 type I (11), 504

Regression line, 90 Regular singular point, 247 Regular SturmLiouville problem,

properties of, 674 Relative error, 71 Relative growth rate, 82 Repeller, 42,578 Residue(s):

definition of, 873 evaluation of integrals by, 876. at a pole of order n, 874 at a simple pole, 874, 875

Residue theorem, 876 Resonance:

curve, 162 frequency, 156, 162 pure, 156

Response: impulse, 227 of a series circuit, 77 of a system, 27,77, 147,603 zero-input, 202 - zero-state, 202

Rest point, 616 Ricatti's differential equation, 67 Riemann mapping theorem, 908 Riemann sum, 535 Right-hand rule, 3 17 Robin condition, 694 Rocket motion, 174 Rodrigues' formula, 266 Roll, 350 Root mean square, 661 Root test, 855 Rotating string, 167-168 Rotation in the z-plane, 891 Round-off error, 269 Row-echelon form, 355 Row equivalent matrices, 355 Row reduction, 377 Row space, 364 Row vectors, 343, 363 Runge-Kutta-Fehlberg algorithm, Runge-Kutta methods, 273-274

fist-order, 274 fourth-order, 274 second-order, 274 for systems, 283

Index

Sawtooth function, 223 Scalar matrix, 347 Scaling, 414 T

Schwarz-Christoffel transformation, 786, 909 Second moments, 508,537 Second-order boundary-value problems, 163 Second-order chemical reactions, 24,85-86 Second-order Runge-Kutta method, 274 Second translation theorem, 208

alternative form of, 209 inverse form of, 209

Self-adjoint form, 676 Semistable critical point, 42 Separable differential equations:

ordinary, 44-45 partial, 688

Separation constant, 689 Series circuits, 25,77, 157 Series (infinite), 853

absolutely convergent, 236, 854 convergent, 236 Fourier, 658,663 Fourier-Bessel, 680 Fourier-Legendre, 682,683 geometric, 853 power, 236,855 solutions of ordinary differential equations, 238 test for convergence, 854

Shaft through the earth, 3 1 Shooting method, 289 Shroud of Turin, 80 Sifting property, 226 Simple closed curve, 486, 833 Simple harmonic electrical vibrations, 157 Simple harmonic motion, 147 Simple pendulum, 172 Simple pole, 870 Simply connected:

domain, 833 region, 497 Simply supported beam, 164 Sine integral function, 58 Sine series in two variables, 716 Single step numerical method, 278 Singular matrix, 3 8 1 Singular point of a complex function:

definition of, 863 essential, 870-871 isolated, 863 nonisolated, 863 pole, 870-871 removable, 870-871 Singular point of a linear differential equation, 54,239,247 irreDdar, 247 regular, 247 Singular solution, 12 Singular Stunn-Liouville problem, 675-676 Sink, 484, 845

Index Index n

&' SIR model, 97 Stationary point, 38,604 Tables: Skew-symmetric matrix, 380 Steady-state: of conformal mappings, 937 Sliding bead, 628-629 current, 78 of derivatives, 929

fluid flow, 920 of integrals, 930 solution, 155, 158 of Laplace transforms, 933

Smooth curve, 450,486 term, 155 Tacoma Narrows suspension bridge, collapse of, 2 Smooth surface, 521 temperature, 694 Tangent plane to a surface:

Step size, 69 definition of, 478 Stochastic matrix, 400 equation of, 478

Solution curve, 9 Stokes' theorem, 529 vector equation of, 478 Solution of an ordinary differential equation: Stream function, 920 Tangential component of acceleration, 460-461

definition of, 8 Streamlines, 802,920 Taylor series, 144, 237, 858 existence and uniqueness of, 17-18, 102 Streamlining, 921 Taylor's theorem, 859

String of length n, 430 Telegraph equation, 696 Sturrn-Liouville problem: Terminal velocity of a falling body, 43,81,89

orthogonality of solutions, 674-675 Thermal diffusivity, 693 properties of, 674 Three-dimensional Laplacian, 692 regular, 673-674 Torque, 3 17 singular, 675-676 Tomcelli's law, 24

Submatrix, 351 Total differential, 59 Subspace, 334 Trace:

Solution of a linear partial differential equation: Substitutions in differential equations, 64 of a matrix, 609 definition of, 688 Successive mappings, 892 of a plane, 327

Superposition principle: Tracer, 439 for Dirichlet's problem for a rectangular plate, 706-707 i Tractrix, 3 1

Solution of a system of algebraic equations: for linear ordinary differential equations, 105, 11 1 Trajectories, 569 definition of, 352 for linear partial differential equations, 690 orthogonal, 100 number of parameters in a solution, 366 for systems of linear differential equations, 570 Transfer coefficients, 440

Solution of a system of differential equations: Surface: Transfer function, 202 definition of, 12, 181,569 area of, 522 Transfer matrix, 440

differential of, 523 I Transform pair, 750,757

integral, 523-524 I Transient: Source, 484, 845 mass of, 524 ! solution, 155, 158 Space curve, 448 orientable, 525 term, 78, 155

Symmetric matrix, 348,404-405,422 I Translation theorems for Laplace transform, 204, 208 eigenvalues of, 405 Translation in the z-plane, 891 orthogonality of eigenvectors, 406 Transpose of a matrix, 376

Sparse matrix, 771 Syndrome, 433 properties of, 346 Spherical Bessel functions, 261 Systematic elimination, 18 1 Traveling waves, 704 Spherical coordinates, 541 Systems of linear algebraic equations, 35 1 Tree diagrams, 468

Laplacian in, 730 consistent, 352 Triangle inequality, 3 16,792 Spiral points, 613 homogeneous, 352 Triangular wave, 223 Spread of a disease, 23, 28 ill-conditioned, 394 Tridiagonal matrix, 777 Square errors, sum of, 437 inconsistent, 352 Triple integral: Square norm, 653 nonhomogeneous, 352 definition of, 534-535 Square wave, 223 overdetermined, 361 in cylindrical coordinates, 540 Stability criteria: undetermined, 361 evaluation of, 535

for first-order autonomous equations, 619 Systems of linear algebraic equations, methods for solving: I in spherical coordinates, 542 for linear systems, 615 Cramer's rule, 391-392 1 Triple scalar product, 3 19 for plane autonomous systems, 621 Gaussian elimination, 355 Triple vector product, 320

Stability of numerical methods, 279 Gauss-Jordan elimination, 355 Triply connected domain, 833 Stagnation point in a flow, 924 Inverse of a matrix, 387 Trivial solution, 9 Staircase function, 213 Using LU-factorization, 362 Trivial vector space, 332 Standard basis, 335 Systems of linear differential equations, methods for solving: Truncation error: Standard forms for linear differential equations, 51,132,135 differential operators, use of, 181 for Euler's method, 269 Standing waves, 702,726 Laplace transforms, 227 I global, 270 State of a system, 27 matrices, use of, 575 for improved Euler's method, 272 State variables, 27 systematic elimination, use of, 181 for Runge-Kutta method, 276 !

I

Tsunami model, 90 Twisted shaft, 713 Two-dimensional heat equation, 715 Two-dimensional Laplacian, 692,720 Two-dimensional wave equation, 715 Two-point boundary-value problem, 103, 163 Type I (11) region, 504

Underdamped circuit, 157 Underdamped motion, 152 Underdetermined system, 361 Undetermined coefficients, 123 Uniqueness theorems, 18, 102,570 Unit impulse, 224 Unit step function, 207 Unit tangent, 459 Unit vector, 301, 331 Unstable critical point, 42, Unstable numerical method, 279

Van der Pol's differential equation, 641 Variable mass and Newton's second law, 30, 175 Variables, separable, 44, 688 Variation of parameters:

for linear ordinary differential equations, 52, 132 for systems of linear differential equations, 591

Vector(s): addition, 298-299, 306 angle between, 3 1 1 components of, 299, in a coordinate plane, 299 coplanar, 321 cross product of, 317,319 difference of, 298-299, differential operator, 470 direction, 323 direction angles of, 3 12 direction cosines of, 312 dot product of, 309, 310, 331 equal, 299,306 equation for a line, 323 equation for a plane, 325 fields, 480 free, 298 function, 448 horizontal component of, 301 initial point of, 298 inner product, 309 length of, 300 linear combination of, 301 magnitude of, 300, multiplication by scalars, 298,299, 306 in n-space, 33 1 negative of, 298, 299,306 norm of, 300 normal to a plane, 326

I I i 1 I

A-96

Vector@) (continued) \7;hrQ+;-, LA-- ?A? q1 A

orthogonal, 3 11,33 1 parallel, 298, 318 position, 299, projection, 3 13 properties of, 300, right-hand rule, 304, scalar multiple of, 298-299, scalar product, 309 as solutions of systems of linear differential equations, 569 subtraction of, 299

r ~ v r a n l l & ucall, / U J , I 1 4

Vibrating string, 693-694, 700 Vibrations, springlmass systems, 147-156, 170-172, 180,

184-185 Virga, 32 Viscous damping, 26 Volterra integral equation, 219 Volterra's principle, 633 Volume:

using double integrals, 503 nC nnmli-i--:--a qnn WL a yaCULcLGpLpCU, J L V

sum of, 298 using triple integrals, 536 terminal point of, 299 Von Karman, Theodore, 2 three-dimensional, 306 Von Karman vortice -

I triple scalar product, 3 19 \T.-,.+~v O ~ A

triple vectorproduct, 320 two-dimensional, 299 unit, 301, 331 valued-function, 448 vertical component of, 301 zero, 298, 300,306

S, L " WILL.',) 7L.v

Vortex point, 616

( v a v L. u y u a u u l l .

derivation of one-dimensional equation, 693-694 nna rl;-=..-:---l A L ~ Lnn '96 vrrb ullcllalullcu, ~ O Y , OYL, o

Vector fields, 480, 603 solution of, 700 and analyticity, 918 two-dimensional, 715, 915

I I conservative, 499,501 Weight, 26 I curl of, 482 \XTa;nh+ &.-*+:-- qqr r r r 1;

divergence of, 483 flux of, 482

r r bl&ll~ ~ U l l L u u l I , L L U , OJJ

Word, 430

finite dimensional, 336 infinite dimensional. 336 real, 332 zero, 332

I Velocity, 454

t field, 458

I potential, complex, 920

Verhulst, P. F., 83 I

I

LUL

UL Lul1 lp lc .x cosine ana sine, 819-820 of complex hyperbolic cosine and sine, 821 ,.c r..--.:-- 07- ul a LUI~LUUII, o I L

of order n, 872 Va-, -+-+A -------. nnn LGlU->LaLC ICbpuIISe , L v L

Zero vector, 298, 300, 306, 334 *

Zero vector space, 332 z-plane, 792

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