一、Syllabus:
I. Introduction to linear spaces
1. linear spaces: linear combination, spanning set, linear dependence,
linear independence, dimension, basis
2. metric spaces: Cauchy sequence, convergent sequence, completeness,
fixed point, contraction mapping, fixed point theorem
3. normed spaces, natural metric, l-p norm, L-p norm
4. inner product spaces: natural norm, Schwartz inequality, Gram-Schmidt
orthgonlization, orthonormal basis, dual bases, adjoint operator, self-
adjoint, eigenvalue problem, eigenexpansion, sets of measure zero
II. Cartesian Tensors
1. Orthonormal Base Vectors
2. Transformation rule of Vectors
3. Scalar, Vector, Pseudo Vector, Pseudo Scalar
4. Dyads, Dyadics, and Tensors
5. Transformation rule of Tensors
6. Quotient Tests
7. Isotropic Tensors

III. Ordinary Differential Equations
1. Initial-Value Problem
2. Existence and Uniqueness Theory
3. System of 1st order ODE’s (const. coefficients)
4. Second-Order ODE
5. Adjoint Operators
6. Green`s Functions and Modified Green`s Function
7. Sturm-Liouville Theory

IV. Partial Differential Equation
1. Introduction
2. Classifications
3. Green`s Function & Integral Representation
4. Other Methods of Solution
5. Maximum-Minimum Principle

二、Prerequisite:
Calculus; Engineering Math (I & II), or Advanced Calculus