MATH 4661 – Content Standards
Title: ADVANCED CALCULUS I
(We propose the title be changed to Mathematical Analysis I.
Catalog Description: The real number system, basic topology of metric spaces, sequences and series, limits and continuity.
Prerequisite: Math 3435. Corequisite: Math 4435.
Goals: Students will study the fundamental concepts of and results in Mathematical Analysis. They will gain a deeper insight into the subject including sequences and series, limits and continuity in the setting of metric spaces. Students will be introduced to the basic proof techniques in Mathematical Analysis. They will be required to present proofs.
CS 1. The Real Number System.
Students will demonstrate an understanding of the axiomatic structure of the real number system. This includes notions such as countable and uncountable sets, completeness and ordering principles, and Cantor sets.
CS2. Topology of Metric Spaces.
Students will exhibit knowledge of metric properties and topological concepts including open and closed sets, compact and connected sets in the context of metric spaces.
CS3. Sequences.
Students will understand and be able to use various concepts regarding sequences such as the limit of a sequence, subsequences, Cauchy sequences, comparison theorems, and the Bolzano-Weierstrass Theorem.
CS4. Series.
Students will demonstrate an understanding of the main theorem on convergence of series, and notions like positive series, absolutely convergent series, alternating series, and power series. They should be able to apply various criteria for convergence of a series.
CS5. Limits and Continuity.
Students will be familiar with the rigorous epsilon and delta treatment of limits of functions between metric spaces and different characterizations of continuity. They will understand the notion of uniform continuity, will know the properties of continuous function on compact and connected sets, and will be able to classify discontinuities of real functions.
CS6. Mathematical Proofs.
All results in this course will be rigorously proven. Students will develop an ability to read, understand, and reproduce proofs in this course.