Methods of Applied Mathematics

by Hildebrand, Francis B.

• ISBN13: 9780486670027
• ISBN10: 0486670023
• Edition: 2nd
• Format: Paperback
• Copyright: 1992-03-27
• Publisher: Dover Publications

Summary

This invaluable book offers engineers and physicists working knowledge of a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, but nevertheless extremely useful when applied to typical problems in many different fields. It deals principally with linear algebraic equations, quadratic and Hermitian forms, operations with vectors and matrices, the calculus of variations, and the formulations and theory of linear integral equations. Annotated problems and exercises accompany each chapter.

Table of Contents

Matrices and Linear Equations
Introduction	p. 1
Linear equations. The Gauss-Jordan reduction	p. 1
Matrices	p. 4
Determinants. Cramer's rule	p. 10
Special matrices	p. 13
The inverse matrix	p. 16
Rank of a matrix	p. 18
Elementary operations	p. 19
Solvability of sets of linear equations	p. 21
Linear vector space	p. 23
Linear equations and vector space	p. 27
Characteristic-value problems	p. 30
Orthogonalization of vector sets	p. 34
Quadratic forms	p. 36
A numerical example	p. 39
Equivalent matrices and transformations	p. 41
Hermitian matrices	p. 42
Multiple characteristic numbers of symmetric matrices	p. 45
Definite forms	p. 47
Discriminants and invariants	p. 50
Coordinate transformations	p. 54
Functions of symmetric matrices	p. 57
Numerical solution of characteristic-value problems	p. 62
Additional techniques	p. 65
Generalized characteristic-value problems	p. 69
Characteristic numbers of nonsymmetric matrices	p. 75
A physical application	p. 78
Function space	p. 81
Sturm-Liouville problems	p. 88
References	p. 93
Problems	p. 93
Calculus of Variations and Applications	p. 119
Maxima and minima	p. 119
The simplest case	p. 123
Illustrative examples	p. 126
Natural boundary conditions and transition conditions	p. 128
The variational notation	p. 131
The more general case	p. 135
Constraints and Lagrange multipliers	p. 139
Variable end points	p. 144
Sturm-Liouville problems	p. 145
Hamilton's principle	p. 148
Lagrange's equations	p. 151
Generalized dynamical entities	p. 155
Constraints in dynamical systems	p. 160
Small vibrations about equilibrium. Normal coordinates	p. 165
Numerical example	p. 170
Variational problems for deformable bodies	p. 172
Useful transformations	p. 178
The variational problem for the elastic plate	p. 179
The Rayleigh-Ritz method	p. 181
A semidirect method	p. 190
References	p. 192
Problems	p. 193
Integral Equations
Introduction	p. 222
Relations between differential and integral equations	p. 225
The Green's function	p. 228
Alternative definition of the Green's function	p. 235
Linear equations in cause and effect. The influence function	p. 242
Fredholm equations with separable kernels	p. 246
Illustrative example	p. 248
Hilbert-Schmidt theory	p. 251
Iterative methods for solving equations of the second kind	p. 259
The Neumann series	p. 266
Fredholm theory	p. 269
Singular integral equations	p. 271
Special devices	p. 274
Iterative approximations to characteristic functions	p. 278
Approximation of Fredholm equations by sets of algebraic equations	p. 279
Approximate methods of undetermined coefficients	p. 283
The method of collocation	p. 284
The method of weighting functions	p. 286
The method of least squares	p. 286
Approximation of the kernel	p. 292
References	p. 294
Problems	p. 294
Appendix: The Crout Method for Solving Sets of Linear Algebraic Equations	p. 339
A. The procedure	p. 339
B. A numerical example	p. 342
C. Application to tridiagonal systems	p. 344
Answers to Problems	p. 347
Index	p. 357
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