| Ch. 1 | Matrices and Linear Equations | |
| 1.1 | Introduction | 1 |
| 1.2 | Linear equations. The Gauss-Jordan reduction | 1 |
| 1.3 | Matrices | 4 |
| 1.4 | Determinants. Cramer's rule | 10 |
| 1.5 | Special matrices | 13 |
| 1.6 | The inverse matrix | 16 |
| 1.7 | Rank of a matrix | 18 |
| 1.8 | Elementary operations | 19 |
| 1.9 | Solvability of sets of linear equations | 21 |
| 1.10 | Linear vector space | 23 |
| 1.11 | Linear equations and vector space | 27 |
| 1.12 | Characteristic-value problems | 30 |
| 1.13 | Orthogonalization of vector sets | 34 |
| 1.14 | Quadratic forms | 36 |
| 1.15 | A numerical example | 39 |
| 1.16 | Equivalent matrices and transformations | 41 |
| 1.17 | Hermitian matrices | 42 |
| 1.18 | Multiple characteristic numbers of symmetric matrices | 45 |
| 1.19 | Definite forms | 47 |
| 1.20 | Discriminants and invariants | 50 |
| 1.21 | Coordinate transformations | 54 |
| 1.22 | Functions of symmetric matrices | 57 |
| 1.23 | Numerical solution of characteristic-value problems | 62 |
| 1.24 | Additional techniques | 65 |
| 1.25 | Generalized characteristic-value problems | 69 |
| 1.26 | Characteristic numbers of nonsymmetric matrices | 75 |
| 1.27 | A physical application | 78 |
| 1.28 | Function space | 81 |
| 1.29 | Sturm-Liouville problems | 88 |
| References | 93 |
| Problems | 93 |
| Ch. 2 | Calculus of Variations and Applications | 119 |
| 2.1 | Maxima and minima | 119 |
| 2.2 | The simplest case | 123 |
| 2.3 | Illustrative examples | 126 |
| 2.4 | Natural boundary conditions and transition conditions | 128 |
| 2.5 | The variational notation | 131 |
| 2.6 | The more general case | 135 |
| 2.7 | Constraints and Lagrange multipliers | 139 |
| 2.8 | Variable end points | 144 |
| 2.9 | Sturm-Liouville problems | 145 |
| 2.10 | Hamilton's principle | 148 |
| 2.11 | Lagrange's equations | 151 |
| 2.12 | Generalized dynamical entities | 155 |
| 2.13 | Constraints in dynamical systems | 160 |
| 2.14 | Small vibrations about equilibrium. Normal coordinates | 165 |
| 2.15 | Numerical example | 170 |
| 2.16 | Variational problems for deformable bodies | 172 |
| 2.17 | Useful transformations | 178 |
| 2.18 | The variational problem for the elastic plate | 179 |
| 2.19 | The Rayleigh-Ritz method | 181 |
| 2.20 | A semidirect method | 190 |
| References | 192 |
| Problems | 193 |
| Ch. 3 | Integral Equations | |
| 3.1 | Introduction | 222 |
| 3.2 | Relations between differential and integral equations | 225 |
| 3.3 | The Green's function | 228 |
| 3.4 | Alternative definition of the Green's function | 235 |
| 3.5 | Linear equations in cause and effect. The influence function | 242 |
| 3.6 | Fredholm equations with separable kernels | 246 |
| 3.7 | Illustrative example | 248 |
| 3.8 | Hilbert-Schmidt theory | 251 |
| 3.9 | Iterative methods for solving equations of the second kind | 259 |
| 3.10 | The Neumann series | 266 |
| 3.11 | Fredholm theory | 269 |
| 3.12 | Singular integral equations | 271 |
| 3.13 | Special devices | 274 |
| 3.14 | Iterative approximations to characteristic functions | 278 |
| 3.15 | Approximation of Fredholm equations by sets of algebraic equations | 279 |
| 3.16 | Approximate methods of undetermined coefficients | 283 |
| 3.17 | The method of collocation | 284 |
| 3.18 | The method of weighting functions | 286 |
| 3.19 | The method of least squares | 286 |
| 3.20 | Approximation of the kernel | 292 |
| References | 294 |
| Problems | 294 |
| Appendix: The Crout Method for Solving Sets of Linear Algebraic Equations | 339 |
| A. The procedure | 339 |
| B. A numerical example | 342 |
| C. Application to tridiagonal systems | 344 |
| Answers to Problems | 347 |
| Index | 357 |
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