Advanced Calculus

Series
Addison-Wesley
Author
Gerald B. Folland  
Publisher
Pearson
Cover
Softcover
Edition
1
Language
English
Total pages
480
Pub.-date
December 2001
ISBN13
9780130652652
ISBN
0130652652
Related Titles


Product detail

Title no longer available

Description

For undergraduate courses in Advanced Calculus and Real Analysis.

This text presents a unified view of calculus in which theory and practice reinforce each other. It covers the theory and applications of derivatives (mostly partial), integrals, (mostly multiple or improper), and infinite series (mostly of functions rather than of numbers), at a deeper level than is found in the standard advanced calculus books.

Features

  • Single and Multivariable Analysis equally balanced.
  • A focus on calculus itself and its applications-Rather than on the foundations of analysis.
    • Provides students with an integrated treatment of the conceptual and computational aspects of the subject-not found in recent texts.

  • Numerous worked-out examples and exercises throughout-From routine calculations to theoretical arguments, or both.
    • Enables students to explore a broad range of items from straightforward calculations to theoretical issues.

  • Well-organized, modern treatment.
    • Uses modern notation and terminology to offer students an up-to-date presentation of the subject.

  • A chapter on Fourier analysis.
    • Provides students with an important and useful application.

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.


Instructor Resources