Advanced Calculus / Edition 1

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Overview

This book presents a unified view of calculus in which theory and practice reinforces each other. It is about 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 calculus books. Chapter topics cover: Setting the Stage, Differential Calculus, The Implicit Function Theorem and Its Applications, Integral Calculus, Line and Surface Integrals—Vector Analysis, Infinite Series, Functions Defined by Series and Integrals, and Fourier Series. For individuals with a sound knowledge of the mechanics of one-variable calculus and an acquaintance with linear algebra.

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Editorial Reviews

From The Critics
This book treats partial derivatives, multiple integrals, and infinite series of functions at a deeper level than is found in most standard calculus texts, emphasizing intuitive understanding rather than formal proofs and technical definitions. Technical definitions are presented with informal arguments and ideas, sometimes involving infinitesimals, that provide more insight than a more rigorous approach. Worked examples and exercises range from routine calculations to theoretical arguments. A sound knowledge of the mechanics of one-variable calculus is required, and some experience with infinite series and partial derivatives is helpful. The author teaches mathematics at the University of Washington. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780130652652
  • Publisher: Pearson
  • Publication date: 12/28/2001
  • Series: Featured Titles for Advanced Calculus Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 480
  • Sales rank: 905,624
  • Product dimensions: 7.00 (w) x 9.00 (h) x 1.00 (d)

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.

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  • Anonymous

    Posted September 1, 2003

    one of the best

    A truly nice text on advanced calculus/math analysis. The theorems are presented in a well structured way with complete proofs. The technical proofs should be included in the main text though; the author collects them in an appendix. Also, some important theorems (such as Dini's Theorem) are not even mentioned; these can be found in other books. Overall, a nice book, and very good exercises.

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