MATH 4662– Content Standards
Title: ADVANCED CALCULUS II (We proposed title be changed to MATH ANALYSIS II).
Catalog Description: The real number system, basic topology of metric spaces, sequences and series, limits and continuity.
Catalog Description: Differentiation of real functions, Riemann integrals, sequences and series of functions, differentiation and integration of functions of several variables.
Prerequisite: Math 4661.
Goals: This courseis a continuation of Math 4661and is aimed at presenting further
fundamental results in Mathematical Analysis in the area of differentiation, Riemann integration, sequences and series of functions, and functions of several variables. Students will be introduced to new proof techniques in analysis.
CS 1. Differentiation.
Students will understand the concept of derivative of a real function and different Mean-Value Theorems and their consequences. They will also understand and be able to apply L’Hospital’s Rule and various theorems involving derivates of higher order.
CS2. Integrability.
Students will demonstrate an understanding of the rigorous treatment of the Riemann Integral based on Riemann and Darboux sums, the properties of the integral, and the Fundamental Theorem of Calculus.
CS3. Sequences and Series of Functions.
Students will exhibit understanding of the concepts of pointwise and uniform convergence of sequences of functions and uniform convergence of series of functions. They will be able to apply theorems on uniform convergence of sequences of continuous, differentiable, and integrable functions as well as power and Taylor series, respectively analytic functions.
CS4. Functions of Several Variables.
Students will understand the generalization of the concept of derivative and Riemann integral to several variables.
CS5. Mathematical Proofs.
All results in this course will be presented with proofs. Students will further develop their ability to read, understand, and reproduce proofs in the area of Mathematical Analysis. They will become familiar with various new proof techniques.