9780486670027

Methods of Applied Mathematics

by
  • ISBN13:

    9780486670027

  • ISBN10:

    0486670023

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 1992-03-27
  • Publisher: Dover Publications

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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
Introductionp. 1
Linear equations. The Gauss-Jordan reductionp. 1
Matricesp. 4
Determinants. Cramer's rulep. 10
Special matricesp. 13
The inverse matrixp. 16
Rank of a matrixp. 18
Elementary operationsp. 19
Solvability of sets of linear equationsp. 21
Linear vector spacep. 23
Linear equations and vector spacep. 27
Characteristic-value problemsp. 30
Orthogonalization of vector setsp. 34
Quadratic formsp. 36
A numerical examplep. 39
Equivalent matrices and transformationsp. 41
Hermitian matricesp. 42
Multiple characteristic numbers of symmetric matricesp. 45
Definite formsp. 47
Discriminants and invariantsp. 50
Coordinate transformationsp. 54
Functions of symmetric matricesp. 57
Numerical solution of characteristic-value problemsp. 62
Additional techniquesp. 65
Generalized characteristic-value problemsp. 69
Characteristic numbers of nonsymmetric matricesp. 75
A physical applicationp. 78
Function spacep. 81
Sturm-Liouville problemsp. 88
Referencesp. 93
Problemsp. 93
Calculus of Variations and Applicationsp. 119
Maxima and minimap. 119
The simplest casep. 123
Illustrative examplesp. 126
Natural boundary conditions and transition conditionsp. 128
The variational notationp. 131
The more general casep. 135
Constraints and Lagrange multipliersp. 139
Variable end pointsp. 144
Sturm-Liouville problemsp. 145
Hamilton's principlep. 148
Lagrange's equationsp. 151
Generalized dynamical entitiesp. 155
Constraints in dynamical systemsp. 160
Small vibrations about equilibrium. Normal coordinatesp. 165
Numerical examplep. 170
Variational problems for deformable bodiesp. 172
Useful transformationsp. 178
The variational problem for the elastic platep. 179
The Rayleigh-Ritz methodp. 181
A semidirect methodp. 190
Referencesp. 192
Problemsp. 193
Integral Equations
Introductionp. 222
Relations between differential and integral equationsp. 225
The Green's functionp. 228
Alternative definition of the Green's functionp. 235
Linear equations in cause and effect. The influence functionp. 242
Fredholm equations with separable kernelsp. 246
Illustrative examplep. 248
Hilbert-Schmidt theoryp. 251
Iterative methods for solving equations of the second kindp. 259
The Neumann seriesp. 266
Fredholm theoryp. 269
Singular integral equationsp. 271
Special devicesp. 274
Iterative approximations to characteristic functionsp. 278
Approximation of Fredholm equations by sets of algebraic equationsp. 279
Approximate methods of undetermined coefficientsp. 283
The method of collocationp. 284
The method of weighting functionsp. 286
The method of least squaresp. 286
Approximation of the kernelp. 292
Referencesp. 294
Problemsp. 294
Appendix: The Crout Method for Solving Sets of Linear Algebraic Equationsp. 339
A. The procedurep. 339
B. A numerical examplep. 342
C. Application to tridiagonal systemsp. 344
Answers to Problemsp. 347
Indexp. 357
Table of Contents provided by Blackwell. All Rights Reserved.

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