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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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