Introduction to applied mathematics

Front cover......Page 1
Title page......Page 3
Date-line......Page 4
CONTENTS......Page 5
Preface......Page 7
Title......Page 11
CHAPTER 1 IN OUTLINE: SYMMETRIC LINEAR SYSTEMS......Page 12
CHAPTER 2 IN OUTLINE: EQUILIBRIUM EQUATIONS......Page 166
CHAPTER 3 IN OUTLINE: EQUILIBRIUM IN THE CONTINUOUS CASE......Page 274
CHAPTER 4 IN OUTLINE: ANALYTICAL METHODS......Page 378
CHAPTER 5 IN OUTLINE: NUMERICAL METHODS......Page 482
CHAPTER 6 IN OUTLINE: INITIAL-VALUE PROBLEMS......Page 618
CHAPTER 7 IN OUTLINE: NETWORK FLOWS AND COMBINATORICS......Page 675
CHAPTER 8 IN OUTLINE: OPTIMIZATION......Page 676
1.1 Introduction......Page 13
1.2 Gaussian Elimination......Page 16
1.3 Positive Definite Matrices and $A = LDL^T$......Page 27
1.4 Minimum Principles......Page 44
1.5 Eigenvalues and Dynamical Systems......Page 59
1.6 A Review of Matrix Theory......Page 80
2.1 A Framework for the Applications......Page 99
2.2 Constraints and Lagrange Multipliers......Page 108
2.3 Electrical Networks......Page 122
2.4 Structures in Equilibrium......Page 135
2.5 Least Squares Estimation and the Kalman Filter......Page 149
3.1 Introduction: One-dimensional Problems......Page 167
3.2 Differential Equations of Equilibrium......Page 178
3.3 Laplace's Equation and Potential Flow......Page 194
3.4 Vector Calculus in Three Dimensions......Page 211
3.5 Equilibrium of Fluids and Solids......Page 232
3.6 Calculus of Variations......Page 254
4.1 Fourier Series and Orthogonal Expansions......Page 275
4.2 Discrete Fourier Series and Convolution......Page 302
4.3 Fourier Integrals......Page 321
4.4 Complex Variables and Conformal Mapping......Page 342
4.5 Complex Integration......Page 364
5.1 Linear and Nonlinear Equations......Page 379
5.2 Orthogonalization and Eigenvalue Problems......Page 394
5.3 Semi-direct and Iterative Methods......Page 415
5.4 The Finite Element Method......Page 440
5.5 The Fast Fourier Transform......Page 460
6.1 Ordinary Differential Equations......Page 483
6.2 Stability and the Phase Plane and Chaos......Page 504
6.3 The Laplace Transform and the $z$-Transform......Page 525
6.4 The Heat Equation vs. the Wave Equation......Page 548
6.5 Difference Methods for Initial-Value Problems......Page 574
6.6 Nonlinear Conservation Laws......Page 599
7.1 Spanning Trees and Shortest Paths......Page 619
7.2 The Marriage Problem......Page 632
7.3 Matching Algorithms......Page 641
7.4 Maximal Flow in a Network......Page 652
7.5 The Transportation Problem......Page 663
8.1 Introduction to Linear Programming......Page 677
8.2 The Simplex Method and Karmarkar's Method......Page 685
8.3 Duality in Linear Programming......Page 704
8.4 Saddle Points (Minimax) and Game Theory......Page 716
8.5 Nonlinear Optimization......Page 730
Appendix: Software for Scientific Computing......Page 747
References and Acknowledgements......Page 750
Solutions to Selected Exercises......Page 753
Index......Page 763
Back cover......Page 784