Methods of Applied Mathematics

CHAPTER ONE Matrices and Linear Equations 1.1. Introduction 1.2. Linear equations. The Gauss-Jordan reduction 1.3. Matrices 1.4. Determinants. Cramer's rule 1.5. Special matrices 1.6. The inverse matrix 1.7. Rank of a matrix 1.8. Elementary operations 1.9. Solvability of sets of linear equations 1.10. Linear vector space 1.11. Linear equations and vector space 1.12. Characteristic-value problems 1.13. Orthogonalization of vector sets 1.14. Quadratic forms 1.15. A numerical example 1.16. Equivalent matrices and transformations 1.17. Hermitian matrices 1.18. Multiple characteristic numbers of symmetric matrices 1.19. Definite forms 1.20. Discriminants and invariants 1.21. Coordinate transformations 1.22. Functions of symmetric matrices 1.23. Numerical solution of characteristic-value problems 1.24. Additional techniques 1.25. Generalized characteristic-value problems 1.26. Characteristic numbers of nonsymmetric matrices 1.27. A physical application 1.28. Function space 1.29. Sturm-Liouville problems References Problems CHAPTER TWO Calculus of Variations and Applications 2.1. Maxima and minima 2.2. The simplest case 2.3. Illustrative examples 2.4. Natural boundary conditions and transition conditions 2.5. The variational notation 2.6. The more general case 2.7. Constraints and Lagrange multipliers 2.8. Variable end points 2.9. Sturm-Liouville problems 2.10. Hamilton's principle 2.11. Lagrange's equations 2.12. Generalized dynamical entities 2.13. Constraints in dynamical systems 2.14. Small vibrations about equilibrium. Normal coordinates 2.15. Numerical example 2.16. Variational problems for deformable bodies 2.17. Useful transformations 2.18. The variational problem for the elastic plate 2.19. The Rayleigh-Ritz method 2.20 A semidirect method References Problems CHAPTER THREE Integral Equations 3.1. Introduction 3.2. Relations between differential and integral equations 3.3. The Green's function 3.4. Alternative definition of the Green's function 3.5. Linear equations in cause and effect. The influence function 3.6. Fredholm equations with separable kernels 3.7. Illustrative example 3.8. Hilbert-Schmidt theory 3.9. Iterative methods for solving equations of the second kind 3.10. The Neumann series 3.11. Fredholm theory 3.12. Singular integral equations 3.13. Special devices 3.14. Iterative approximations to characteristic functions 3.15. Approximation of Fredholm equations by sets of algebraic equations 3.16. Approximate methods of undetermined coefficients 3.17. The method of collocation 3.18. The method of weighting functions 3.19. The method of least squares 3.20. Approximation of the kernel References Problems APPENDIX The Crout Method for Solving Sets of Linear Algebraic Equations A. The procedure B. A numerical example C. Application to tridiagonal system Answers to Problems Index