Advanced Calculus - 02 edition

Folland, Gerald B : University of Washington

1. Setting the Stage

Euclidean Spaces and Vectors
Subsets of Euclidean Space
Limits and Continuity
Sequences
Completeness
Compactness
Connectedness
Uniform Continuity

2. Differential Calculus

Differentiability in One Variable
Differentiability in Several Variables
The Chain Rule
The Mean Value Theorem
Functional Relations and Implicit Functions: A First Look
Higher-Order Partial Derivatives
Taylor's Theorem
Critical Points
Extreme Value Problems
Vector-Valued Functions and Their Derivatives

3. The Implicit Function Theorem and Its Applications

The Implicit Function Theorem
Curves in the Plane
Surfaces and Curves in Space
Transformations and Coordinate Systems
Functional Dependence

4. Integral Calculus

Integration on the Line
Integration in Higher Dimensions
Multiple Integrals and Iterated Integrals
Change of Variables for Multiple Integrals
Functions Defined by Integrals
Improper Integrals
Improper Multiple Integrals
Lebesgue Measure and the Lebesgue Integral

5. Line and Surface Integrals; Vector Analysis

Arc Length and Line Integrals
Green's Theorem
Surface Area and Surface Integrals
Vector Derivatives
The Divergence Theorem
Some Applications to Physics
Stokes's Theorem
Integrating Vector Derivatives
Higher Dimensions and Differential Forms

6. Infinite Series

Definitions and Examples
Series with Nonnegative Terms
Absolute and Conditional Convergence
More Convergence Tests
Double Series; Products of Series

7. Functions Defined by Series and Integrals

Sequences and Series of Functions
Integrals and Derivatives of Sequences and Series
Power Series
The Complex Exponential and Trig Functions
Functions Defined by Improper Integrals
The Gamma Function
Stirling's Formula

8. Fourier Series

Periodic Functions and Fourier Series
Convergence of Fourier Series
Derivatives, Integrals, and Uniform Convergence
Fourier Series on Intervals
Applications to Differential Equations
The Infinite-Dimensional Geometry of Fourier Series
The Isoperimetric Inequality

APPENDICES

A. Summary of Linear Algebra

Vectors
Linear Maps and Matrices
Row Operations and Echelon Forms
Determinants
Linear Independence
Subspaces; Dimension; Rank
Invertibility
Eigenvectors and Eigenvalues

B. Some Technical Proofs

The Heine-Borel Theorem
The Implicit Function Theorem
Approximation by Riemann Sums
Double Integrals and Iterated Integrals
Change of Variables for Multiple Integrals
Improper Multiple Integrals
Green's Theorem and the Divergence Theorem
Answers to Selected Exercises
Bibliography
Index