Advanced Calculus

ISBN-10: 0130652652
ISBN-13: 9780130652652
Edition: 2002
List price: $117.80
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Description: 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  More...

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Book details

List price: $117.80
Copyright year: 2002
Publisher: Prentice Hall PTR
Publication date: 12/21/2001
Binding: Paperback
Pages: 480
Size: 7.25" wide x 9.25" long x 1.00" tall
Weight: 1.892
Language: English

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 IntegralsVector 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.

Preface
Setting the Stage
Euclidean Spaces and Vectors
Subsets of Euclidean Space
Limits and Continuity
Sequences
Completencess
Compactness
Connectedness
Uniform Continuity
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
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
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
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
Infinite Series
Definitions and Examples
Series with Nonnegative Terms
Absolute and Conditional Convergence
More Convergence Tests
Double Series; Products of Series
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
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
Summary of Linear Algebra
Vectors
Linear Maps and Matrices
Row Operations and Echelon Forms
Determinants
Linear Independence
Subspaces; Dimension; Rank
Invertibility
Eigenvectors and Eigenvalues
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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