COURSE NUMBER AND TITLE: MATH 4012 Real Variables II

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of differentiation and integration of functions on n-dimensional Euclidian space. Other topics include the elementary theory of metric spaces, infinite sequences and series, and Fourier series.

PREREQUISITE(S): MATH 2013 and MATH 4011

SUGGESTED TEXT(S): A First Course in Real Analysis, by Protter and Morrey

COURSE OUTLINE:

1. Infinite Sequences and Infinite Series
   • Tests for Convergence and Divergence
   • Series of Positive and Negative Terms; Power Series
   • Uniform Convergence of Sequences
   • Uniform Convergence of Series; Power Series
   • Unordered Sums
   • The Comparison Test for Unordered Sums; Uniform Convergence
   • Multiple sequences and Series
2. Fourier Series
   • Expansions of Periodic Functions
   • Sine Series and Cosine Series; Change of Interval
   • Convergence Theorems
3. Functions Defined by Integrals; Improper Integrals
   • The Derivative of a Function Defined by an Integral; the Leibniz Rule
   • Convergence and Divergence of Improper Integrals
   • The Derivative of Functions Defined by Improper Integrals
   • The Gamma Function
4. The Riemann-Stieltjes Integral and Functions of Bounded Variation
   • Functions of Bounded Variation
   • The Riemann-Stieltjes Integral
5. Contraction Mappings, Newton’s Method, and Differential Equations
   • A Fixed Point Theorem and Newton’s Method
   • Application of the Fixed Point Theorem to Differential Equations
6. Vector Field Theory; the Theorems of Green and Stokes
   • Vector Functions on R1
   • Vector Functions and Fields on RN
   • Line Integrals in RN
   • Green’s Theorem in the Plane
   • Surfaces in R3; Parametric Representation
   • Area of a Surface in R3; Surface Integrals
   • Orientable Surfaces
   • Stokes’ Theorem
   • The Divergence Theorem