1. Introduction
(a) Preliminaries
(b) The exponential function
(c) Differential Equations as Mathematical Models
2. Linear Algebra
(a) Matrices, Determinants and Basics
(b) O(N3), O(N2) and O(N) operations
(c) Orthogonal Matrices
(d) Eigenvalues, Eivenvectors, and Similarity Transform
(e) The normal modes of a vibrating system
(f) Hermitian and skew-Hermitian Matrices
(g) Orthogonal Bases, Projection and Expansions
3. Ordinary Differential Equations
(a) First Order Differential Equations
(b) Second Order Differential Equations
(c) Autonomous systems, Stability, and Phase Plane
(d) Sturm-Liouville Theory
(e) Series Solution and Some Special Functions
(f) Integral Transforms
4. Vector Analysis
(a) Definitions, Elementary Approach
(b) Scalar or Dot Product
(c) Vector or Cross Product
(d) The gradient
(e) The divergence of a vector field
(f) The curl of a vector field
(g) The Laplacian
(h) Differential Vector Operators in Curved Coordinate (Spherical and cylindrical coodinates)
(i) The Gauss Theorem and Poisson Equation
(j) The Stoke theorem
(k) Helmholtz’s Theorem
(l) Successive Application of (Einstein’s notation or Levi-civita symbols)
(m) Maxwell equations and Euler equations
(n) Conservation laws