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Basic Analysis II: Introduction to Real Analysis, Volume II

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Version 2.6. (Newer edition 6 available, ISBN 979-8851945977) The second volume of Basic Analysis, a first course in mathematical analysis. This volume is the second semester material for a year-long sequence for advanced undergraduates or masters level students. This volume started with notes for Math 522 at University of Wisconsin-Madison, and then was heavily revised and modified for teaching Math 4153/5053 at Oklahoma State University. It covers differential calculus in several variables, line integrals, multivariable Riemann integral including a basic case of Green's Theorem, and topics on power series, Arzelà-Ascoli, Stone-Weierstrass, and Fourier Series. See http://www.jirka.org/ra/

Table of Contents (of this volume II):
8. Several Variables and Partial Derivatives
9. One Dimensional Integrals in Several Variables
10. Multivariable Integral
11. Functions as Limits

Author(s): Jirí Lebl
Series: Basic Analysis: Introduction to Real Analysis 02
Edition: 6.0
Publisher: CreateSpace Independent Publishing Platform
Year: 2023

Language: English
Tags: 18.100A; maths; mathematics; math; calculus; Massachusetts Institute of Technology; MIT

Title Page
Introduction
Several Variables and Partial Derivatives
Vector spaces, linear mappings, and convexity
Analysis with vector spaces
The derivative
Continuity and the derivative
Inverse and implicit function theorems
Higher order derivatives
One-dimensional Integrals in Several Variables
Differentiation under the integral
Path integrals
Path independence
Multivariable Integral
Riemann integral over rectangles
Iterated integrals and Fubini theorem
Outer measure and null sets
The set of Riemann integrable functions
Jordan measurable sets
Green's theorem
Change of variables
Functions as Limits
Complex numbers
Swapping limits
Power series and analytic functions
Complex exponential and trigonometric functions
Maximum principle and the fundamental theorem of algebra
Equicontinuity and the Arzelà–Ascoli theorem
The Stone–Weierstrass theorem
Fourier series
Further Reading
Index
List of Notation