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Cover type: Paperback

Edition: 02

Copyright: 2002

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

Edition: 02

Copyright: 2002

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

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**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

Author Bio

**Folland, Gerald B : University of Washington **

Table of Contents

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

Publisher Info

Publisher: Prentice Hall, Inc.

Published: 2002

International: No

Published: 2002

International: No

**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