Weber / Arfken Essentials of Math Methods for Physicists

1. Auflage 2013
ISBN: 978-1-4832-2562-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 960 Seiten, Web PDF

Essentials of Math Methods for Physicists aims to guide the student in learning the mathematical language used by physicists by leading them through worked examples and then practicing problems. The pedagogy is that of introducing concepts, designing and refining methods and practice them repeatedly in physics examples and problems. Geometric and algebraic approaches and methods are included and are more or less emphasized in a variety of settings to accommodate different learning styles of students. Comprised of 19 chapters, this book begins with an introduction to the basic concepts of vector algebra and vector analysis and their application to classical mechanics and electrodynamics. The next chapter deals with the extension of vector algebra and analysis to curved orthogonal coordinates, again with applications from classical mechanics and electrodynamics. These chapters lay the foundations for differential equations, variational calculus, and nonlinear analysisin later discussions. High school algebra of one or two linear equations is also extended to determinants and matrix solutions of general systems of linear equations, eigenvalues and eigenvectors, and linear transformations in real and complex vector spaces. The book also considers probability and statistics as well as special functions and Fourier series. Historical remarks are included that describe some physicists and mathematicians who introduced the ideas and methods that were perfected by later generations to the tools routinely used today. This monograph is intended to help undergraduate students prepare for the level of mathematics expected in more advanced undergraduate physics and engineering courses.

Autoren/Hrsg.

Weber, Hans J.

Arfken, George B.

Inhaltsverzeichnis

1;Front Cover;1
2;Essentials of Math Methods for Physicists;2
3;Table of Contents;4
4;Preface;10
4.1;Text Overview;10
4.2;Pathways through the Material;11
4.3;Problem-Solving Skills;11
4.4;Computer Software;12
4.5;Acknowledgements;12
5;Chapter 1. VECTOR ANALYSIS;14
5.1;1.1 ELEMENTARY APPROACH;14
5.2;1.2 SCALAR OR DOT PRODUCT;26
5.3;1.3 VECTOR OR CROSS PRODUCT;35
5.4;1.4 TRIPLE SCALAR PRODUCT, TRIPLE VECTOR PRODUCT;45
5.5;1.5 GRADIENT, V;51
5.6;1.6 DIVERGENCE, V;62
5.7;1.7 CURL, Vx;65
5.8;1.8 SUCCESSIVE APPLICATIONS OF V;72
5.9;1.9 VECTOR INTEGRATION;77
5.10;1.10 GAUSS' THEOREM;86
5.11;1.11 STOKES' THEOREM;90
5.12;1.12 POTENTIAL THEORY;94
5.13;1.13 GAUSS' LAW, POISSON'S EQUATION;101
5.14;1.14 DIRAC DELTA FUNCTION;107
5.15;ADDITIONAL READINGS;116
6;Chapter 2. VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS;118
6.1;2.1 SPECIAL COORDINATE SYSTEMS: INTRODUCTION;119
6.2;2.2 CIRCULAR CYLINDER COORDINATES;120
6.3;2.3 ORTHOGONAL COORDINATES;137
6.4;2.4 DIFFERENTIAL VECTOR OPERATORS;145
6.5;2.5 SPHERICAL POLAR COORDINATES;150
6.6;2.6 TENSOR ANALYSIS;160
6.7;2.7 CONTRACTION, DIRECT PRODUCT;173
6.8;2.8 QUOTIENT RULE;176
6.9;2.9 DUAL TENSORS;178
6.10;ADDITIONAL READINGS;183
7;Chapter 3. DETERMINANTS AND MATRICES;184
7.1;3.1 DETERMINANTS;184
7.2;3.2 MATRICES;200
7.3;3.3 ORTHOGONAL MATRICES;221
7.4;3.4 HERMITIAN MATRICES, UNITARY MATRICES;234
7.5;3.5 DIAGONALIZATION OF MATRICES;239
7.6;ADDITIONAL READINGS;257
8;Chapter 4. GROUP THEORY;260
8.1;4.1 INTRODUCTION TO GROUP THEORY;260
8.2;4.2 GENERATORS OF CONTINUOUS GROUPS;268
8.3;4.3 ORBITAL ANGULAR MOMENTUM;275
8.4;4.4 HOMOGENEOUS LORENTZ GROUP;280
9;Chapter 5. INFINITE SERIES;290
9.1;5.1 FUNDAMENTAL CONCEPTS;290
9.2;5.2 CONVERGENCE TESTS;295
9.3;5.3 ALTERNATING SERIES;303
9.4;5.4 ALGEBRA OF SERIES;308
9.5;5.5 SERIES OF FUNCTIONS;310
9.6;5.6 TAYLOR'S EXPANSION;315
9.7;5.7 POWER SERIES;325
9.8;5.8 ELLIPTIC INTEGRALS;331
9.9;5.9 BERNOULLI NUMBERS, EULER-MACLAURIN FORMULA;338
9.10;5.10 ASYMPTOTIC SERIES;350
9.11;ADDITIONAL READINGS;353
10;Chapter 6. FUNCTIONS OF A COMPLEX VARIABLE I;354
10.1;6.1 COMPLEX ALGEBRA;356
10.2;6.2 CAUCHY-RIEMANN CONDITIONS;368
10.3;6.3 CAUCHY'S INTEGRAL THEOREM;375
10.4;6.4 CAUCHY'S INTEGRAL FORMULA;381
10.5;6.5 LAURENT EXPANSION;386
10.6;6.6 MAPPING;397
10.7;6.7 CONFORMAL MAPPING;406
10.8;ADDITIONAL READINGS;408
11;Chapter 7. FUNCTIONS OF A COMPLEX VARIABLE II;410
11.1;7.1 SINGULARITIES;410
11.2;7.2 CALCULUS OF RESIDUES;416
11.3;7.3 METHOD OF STEEPEST DESCENTS;439
11.4;ADDITIONAL READINGS;448
12;Chapter 8. DIFFERENTIAL EQUATIONS;450
12.1;8.1 INTRODUCTION;450
12.2;8.2 FIRST ORDER ODEs;451
12.3;8.3 SECOND ORDER ODEs;466
12.4;8.4 SINGULAR POINTS;481
12.5;8.5 SERIES SOLUTIONS–FROBENIUS' METHOD;483
12.6;8.6 A SECOND SOLUTION;496
12.7;8.7 NUMERICAL SOLUTIONS;507
12.8;8.8 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS;513
12.9;8.9 SEPARATION OF VARIABLES;514
12.10;ADDITIONAL READINGS;524
13;Chapter 9. STURM-LIOUVILLE THEORY–ORTHOGONAL FUNCTIONS;526
13.1;9.1 SELF-ADJOINT ODEs;527
13.2;9.2 HERMITIAN OPERATORS;541
13.3;9.3 GRAM-SCHMIDT ORTHOGONALIZATION;548
13.4;9.4 COMPLETENESS OF EIGENFUNCTIONS;555
13.5;ADDITIONAL READINGS;568
14;Chapter 10. THE GAMMA FUNCTION(FACTORIAL FUNCTION);570
14.1;10.1 DEFINITIONS, SIMPLE PROPERTIES;570
14.2;10.2 DIGAMMA AND POLYGAMMA FUNCTIONS;583
14.3;10.3 STIRLING'S SERIES;588
14.4;10.4 THE INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS;593
14.5;ADDITIONAL READINGS;599
15;Chapter 11. LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS;600
15.1;11.1 INTRODUCTION;600
15.2;11.2 RECURRENCE RELATIONS AND SPECIAL PROPERTIES;610
15.3;11.3 ORTHOGONALITY;616
15.4;11.4 ALTERNATE DEFINITIONS OF LEGENDRE POLYNOMIALS;627
15.5;11.5 ASSOCIATED LEGENDRE FUNCTIONS;630
15.6;ADDITIONAL READINGS;638
16;Chapter 12. BESSEL FUNCTIONS;640
16.1;12.1 BESSEL FUNCTIONS OF THE FIRST KIND, Jv(x);640
16.2;12.2 NEUMANN FUNCTIONS, BESSEL FUNCTIONS OF THE SECOND KIND;666
16.3;12.3 ASYMPTOTIC EXPANSIONS;672
16.4;12.4 SPHERICAL BESSEL FUNCTIONS;679
16.5;ADDITIONAL READINGS;694
17;Chapter 13. HERMITE AND LAGUERRE POLYNOMIALS;696
17.1;13.1 HERMITE POLYNOMIALS;696
17.2;13.2 LAGUERRE FUNCTIONS;709
17.3;ADDITIONAL READINGS;720
18;Chapter 14. FOURIER SERIES;722
18.1;14.1 GENERAL PROPERTIES;722
18.2;14.2 ADVANTAGES, USES OF FOURIER SERIES;730
18.3;14.3 COMPLEX FOURIER SERIES;736
18.4;14.4 PROPERTIES OF FOURIER SERIES;741
18.5;ADDITIONAL READINGS;747
19;Chapter 15. INTEGRAL TRANSFORMS;748
19.1;15.1 INTRODUCTION, DEFINITIONS;748
19.2;15.2 FOURIER TRANSFORM;749
19.3;15.3 DEVELOPMENT OF THE INVERSE FOURIER TRANSFORM;753
19.4;15.4 FOURIER TRANSFORMS—INVERSION THEOREM;758
19.5;15.5 FOURIER TRANSFORM OF DERIVATIVES;770
19.6;15.6 CONVOLUTION THEOREM;777
19.7;15.7 MOMENTUM REPRESENTATION;783
19.8;15.8 LAPLACE TRANSFORMS;789
19.9;15.9 LAPLACE TRANSFORM OF DERIVATIVES;797
19.10;15.10 OTHER PROPERTIES;801
19.11;15.11 CONVOLUTION OR FALTUNGS THEOREM;809
19.12;15.12 INVERSE LAPLACE TRANSFORM;813
19.13;ADDITIONAL READINGS;823
20;Chapter 16. PARTIAL DIFFERENTIAL EQUATIONS;826
20.1;16.1 EXAMPLES OF PDEs AND BOUNDARY CONDITIONS;826
20.2;16.2 Heat Flow Or Diffusion PDE;831
20.3;16.3 INHOMOGENEOUS PDE—GREEN'S FUNCTION;840
20.4;ADDITIONAL READINGS;851
21;Chapter 17. PROBABILITY;854
21.1;17.1 DEFINITIONS, SIMPLE PROPERTIES;854
21.2;17.2 RANDOM VARIABLES;861
21.3;17.3 BINOMIAL DISTRIBUTION;875
21.4;17.4 POISSON DISTRIBUTION
;878
21.5;17.5 GAUSS' NORMAL DISTRIBUTION;881
21.6;17.6 STATISTICS;885
21.7;ADDITIONAL READINGS;899
22;Chapter 18. CALCULUS OF VARIATIONS;902
22.1;18.1 A DEPENDENT AND AN INDEPENDENT VARIABLE;903
22.2;18.2 SEVERAL DEPENDENT VARIABLES;914
22.3;18.3 SEVERAL INDEPENDENT VARIABLES;922
22.4;18.4 SEVERAL DEPENDENT AND INDEPENDENT VARIABLES;924
22.5;18.5 LAGRANGIAN MULTIPLIERS: VARIATION WITH CONSTRAINTS;926
22.6;18.6 RAYLEIGH-RITZ VARIATIONAL TECHNIQUE;939
22.6.1;ADDITIONAL READINGS;1003
22.7;ADDITIONAL READINGS;945
23;Chapter 19. NONLINEAR METHODS AND CHAOS;946
23.1;19.1 INTRODUCTION;946
23.2;19.2 THE LOGISTIC MAP;948
23.3;19.3 SENSITIVITY TO INITIAL CONDITIONS AND PARAMETERS;953
23.4;19.4 NONLINEAR DIFFERENTIAL EQUATIONS;957
23.5;ADDITIONAL READINGS;997
24;APPENDIX;999
24.1;BISECTION METHOD;999
25;GENERAL REFERENCES;1006
26;VECTOR CALCULUS;1008
27;CURVED ORTHOGONAL COORDINATES;1008
28;NTEGRALS;1010