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Math 131B: General Course Outline

Catalog Description

131B. Analysis. (4) Lecture, three hours; discussion, one hour. P/NP or letter grading. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of fourth and eighth weeks of instruction, plus reviews for the final exam.

Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.

There is an honors sequence Math 131AH-131BH running parallel to 131A-131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.

Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.

Textbook

(Listed textbook is the standard guideline. Please check the schedule of classes for the official textbook.)

Section 14.8 is the proof of the Weierstrass Approximation Theorem. This should probably be left for the Honors Section.
This is rather difficult, but it introduces summation by parts. Using summation by parts to prove Dirichlet's Test (and hence the Alternating Series Test) is an alternative to Abel's Theorem.
This is a lot, but Sections 17.1 is just a review of linear transformations and 17.2 and 17.3 contain only one theorem.

Outline update: J. Ralston, 4/08

Schedule of Lectures

Lecture Section Topics

1

12.1, 12.2, 12.3

Metric Spaces, Some Point-Set Topology and Relative Topology

2

12.4, 12.5, 13.1

Cauchy Sequences and Completeness, Compact Metric Spaces, Continuous Functions on Metric Spaces

3

13.2, 13.3, 13.4

Continuity on Product, Connected and Compact Metric Spaces

4

14.1, 14.2

Uniform Convergence, Midterm I

5

14.3, 14.5, 14.6, 14.7

Uniform Convergence and Continuity, the "Sup" Norm, Series of Functions, Uniform Convergence in Integration and Differentiation3

6

15.1, 15.2, 15.3, 15.4

Formal Power Series, Real Analytic Functions, Abel's Theorem (Optional)4, Multiplication of Power Series

7

15.5, 15.6, 15.7, 16.1

Exponential and Logarithmic Functions, Trigonometric Functions, Periodic Functions

8

16.2, 16.3

Inner Products on Periodic Functions, Trigonometric Polynomials, Hour Exam II

9

16.4, 16.5, 17.1, 17.2, 17.35

Periodic Convolutions, L2 convergence of Fourier Series and Plancherel's Theorem, Differentiability of Functions of Several Variables

10

17.4, 17.5

The Several Variable Chain Rule, Clairaut's Theorem, Review of Course