Methods of Applied Mathematics / Edition 2

Methods of Applied Mathematics / Edition 2

by Francis Begnaud Hildebrand
     
 

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… See more details below

Overview

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.

Product Details

ISBN-13:
9780135792018
Publisher:
Prentice Hall Professional Technical Reference
Publication date:
03/28/1965

Table of Contents

Ch. 1Matrices and Linear Equations
1.1Introduction1
1.2Linear equations. The Gauss-Jordan reduction1
1.3Matrices4
1.4Determinants. Cramer's rule10
1.5Special matrices13
1.6The inverse matrix16
1.7Rank of a matrix18
1.8Elementary operations19
1.9Solvability of sets of linear equations21
1.10Linear vector space23
1.11Linear equations and vector space27
1.12Characteristic-value problems30
1.13Orthogonalization of vector sets34
1.14Quadratic forms36
1.15A numerical example39
1.16Equivalent matrices and transformations41
1.17Hermitian matrices42
1.18Multiple characteristic numbers of symmetric matrices45
1.19Definite forms47
1.20Discriminants and invariants50
1.21Coordinate transformations54
1.22Functions of symmetric matrices57
1.23Numerical solution of characteristic-value problems62
1.24Additional techniques65
1.25Generalized characteristic-value problems69
1.26Characteristic numbers of nonsymmetric matrices75
1.27A physical application78
1.28Function space81
1.29Sturm-Liouville problems88
References93
Problems93
Ch. 2Calculus of Variations and Applications119
2.1Maxima and minima119
2.2The simplest case123
2.3Illustrative examples126
2.4Natural boundary conditions and transition conditions128
2.5The variational notation131
2.6The more general case135
2.7Constraints and Lagrange multipliers139
2.8Variable end points144
2.9Sturm-Liouville problems145
2.10Hamilton's principle148
2.11Lagrange's equations151
2.12Generalized dynamical entities155
2.13Constraints in dynamical systems160
2.14Small vibrations about equilibrium. Normal coordinates165
2.15Numerical example170
2.16Variational problems for deformable bodies172
2.17Useful transformations178
2.18The variational problem for the elastic plate179
2.19The Rayleigh-Ritz method181
2.20A semidirect method190
References192
Problems193
Ch. 3Integral Equations
3.1Introduction222
3.2Relations between differential and integral equations225
3.3The Green's function228
3.4Alternative definition of the Green's function235
3.5Linear equations in cause and effect. The influence function242
3.6Fredholm equations with separable kernels246
3.7Illustrative example248
3.8Hilbert-Schmidt theory251
3.9Iterative methods for solving equations of the second kind259
3.10The Neumann series266
3.11Fredholm theory269
3.12Singular integral equations271
3.13Special devices274
3.14Iterative approximations to characteristic functions278
3.15Approximation of Fredholm equations by sets of algebraic equations279
3.16Approximate methods of undetermined coefficients283
3.17The method of collocation284
3.18The method of weighting functions286
3.19The method of least squares286
3.20Approximation of the kernel292
References294
Problems294
Appendix: The Crout Method for Solving Sets of Linear Algebraic Equations339
A. The procedure339
B. A numerical example342
C. Application to tridiagonal systems344
Answers to Problems347
Index357

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