MA464

Prerequisite: MA 363 (Advanced Calculus I)

Course Content Outline

Sets and Functions
1. Introduction
2. Geometry
3. Distance
4. Functions
5. Topological ideas
6. Order properties and the LUB property
7. Sequences of points
8. Real sequences
Continuity
1. Introduction
2. Basic definitions
3. Approximation properties and uniform continuity
4. Properties of continuous functions
5. Limits of functions
6. Discontinuities
7. Mean value theorems and L'Hospital's rule
Integration
1. The definite integral
2. Evaluation of definite integrals
3. Taylor's theorem
4. Improper integrals
5. Set functions
Convergence
1. Infinite series
2. Uniform convergence
3. Power series
4. Improper integrals with a parameter
5. The Gamma function
Differentiation
1. Transformations
2. Linear functions and transformations
3. The differential of a function
4. Differentiation of composite functions
5. Differentials of transformations
6. Inverses of functions of one variable
7. Inverses of transformations
8. The implicit function theorems
9. Functional dependence
Applications to Geometry and Analysis
1. Transformations of multiple integrals
2. Curves and arc length
3. Surfaces and surface area
4. Extremal properties of functions of several variables
Differential Geometry and Vector Calculus
1. Integrals over curves and surfaces
2. Differential forms
3. Vector analysis
4. The theorems of Green, Gauss, and Stokes
5. Independence of path, and exact differential forms