MA 464 Course Description

MA464 ANALYSIS II (3 Cr.)

COURSE DESCRIPTION

Prerequisite: MA363 (Analysis I), junior standing or instructor's permission

Offered: On Demand

Course Description
Covers selected topics from Real Analysis and/or Complex Analysis. This course serves as a second course in Analysis.

Infinite series and uniform convergence, power series, transformations, inverse function and implicit function theorems, and calculus of several variables.

Course Content Outline

  1. 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
  2. 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
  3. Integration
    1. The definite integral
    2. Evaluation of definite integrals
    3. Taylor's theorem
    4. Improper integrals
    5. Set functions
  4. Convergence
    1. Infinite series
    2. Uniform convergence
    3. Power series
    4. Improper integrals with a parameter
    5. The Gamma function
  5. 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
  6. 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
  7. 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