# INTRODUCTION TO APPLIED MATHEMATICS

## GILBERT STRANG

### Contents

```1. SYMMETRIC LINEAR SYSTEMS

1.1 Introduction
1.2 Gaussian Elimination
1.3 Positive Definite Matrices and A = LDLT
1.4 Minimum Principles
1.5 Eigenvalues and Dynamical Systems
1.6 A Review of Matrix Theory

2. EQUILIBRIUM EQUATIONS

2.1 A Framework for the Applications
2.2 Constraints and Lagrange Multipliers
2.3 Electrical Networks
2.4 Structures in Equilibrium
2.5 Least Squares Estimation and the Kalman Filter

3. EQUILIBRIUM IN THE CONTINUOUS CASE

3.1 Introduction: One-dimensional Problems
3.2 Differential Equations of Equilibrium
3.3 Laplace's Equation and Potential Flow
3.4 Vector Calculus in Three Dimensions
3.5 Equilibrium of Fluids and Solids
3.6 Calculus of Variations

4. ANALYTICAL METHODS

4.1 Fourier Series and Orthogonal Expansions
4.2 Discrete Fourier Series and Convolution
4.3 Fourier Integrals
4.4 Complex Variables and Conformal Mapping
4.5 Complex Integration

5. NUMERICAL METHODS

5.1 Linear and Nonlinear Equations
5.2 Orthogonalization and Eigenvalue Problems
5.3 Semi-direct and Iterative Methods
5.4 The Finite Element Method
5.5 The Fast Fourier Transform

6. INITIAL-VALUE PROBLEMS

6.1 Ordinary Differential Equations
6.2 Stability and the Phase Plane and Chaos
6.3 The Laplace Transform and the z-Transform
6.4 The Heat Equation vs. the Wave Equation
6.5 Difference Methods for Initial-Value Problems
6.6 Nonlinear Conservation Laws

7. NETWORK FLOWS AND COMBINATORICS

7.1 Spanning Trees and Shortest Paths
7.2 The Marriage Problem
7.3 Matching Algorithms
7.4 Maximal Flow in a Network
7.5 The Transportation Problem

8. OPTIMIZATION

8.1 Introduction to Linear Programming
8.2 The Simplex Method and Karmarkar's Method
8.3 Duality in Linear Programming
8.4 Saddle Points (Minimax) and Game Theory
8.5 Nonlinear Optimization

Appendix: Software for Scientific Computing

References and Acknowledgements

Solutions to Selected Exercises
Index
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