Introduction to Applied Mathematics

Table of Contents

• 1. Symmetric Linear Systems:
• 1.1 Introduction
• 1.2 Gaussian elimination
• 1.3 Positive definite matrices
• 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 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
• 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
• Software for scientific computing
• References and acknowledgements
• Solutions to selected exercises
• Index.
Show more