San Francisco State University
Department of Mathematics
Course Syllabus
MATH 470
Real Analysis II—Several Variables
Prerequisites
MATH 370 with a grade of C or better.
Bulletin Description
Infinite sequences and series of functions, uniform convergence, differentiable
maps from R nn to R m , Inverse Function Theorem, Implicit Function Theorem,
multivariable integration, Fubini’s theorem, Change of Variables Theorem.
Course Objectives
The principal aim of Real Analysis II is for students to learn how to carry out a
rigorous analysis of the convergence of infinite series of functions, properties of
differentiable mappings, Inverse Function and the Implicit Function Theorem.
Students learn how to apply the Change of Variables and Fubini’s theorem in
multivariable Riemann integration. Students learn to solve problems using the
concepts of analysis. They present their solutions as rigorous proofs written in
correct mathematical English. Students will be able to devise, organize and
present brief (half-page) solutions based on definitions and theorems of
analysis.
Students who complete this course should be able to:
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Determine the properties of uniformly convergent sequences and series of
functions.
Determine the properties of differentiable functions of several variables
including the chain rule, product rule, mean value theorem.
Prove and apply the Inverse and Implicit Function Theorem.
Apply Fubini’s theorem and the Change of Variables Theorem in
multivariable integration.
Evaluation of Students
Students will be graded on their ability to devise, organize and present in
correct mathematical English rigorous solutions to problems. While instructors
may design their own methods of evaluating student performance these
methods must include in-class examinations, graded homework assignments
and a final exam.
Course Outline
Topics
Infinite sequences and series of functions: uniform
convergence.
Differentiable maps from R nn to R m : The derivative matrix,
Chain Rule, Mean Value Theorem
Inverse Function Theorem; Implicit Function Theorem
Integrable Functions on R nn : Fubini’s Theorem, Change of
Variables
Number of
Weeks
3
4
3
4
Textbooks and Software
Advanced Calculus, 3rd Edition by R. Creighton Buck.
Elementary Classical Analysis, 2nd Edition, by J. Marsden & M. Hoffman.
Submitted by: David Ellis
Date: 6 Sept 2005