Advanced Calculus

ISBN-10: 0821847910

ISBN-13: 9780821847916

Edition: 2nd 2009 (Revised)

List price: $87.00
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Description:

Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables. Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material. Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis.
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Book details

List price: $87.00
Edition: 2nd
Copyright year: 2009
Publisher: American Mathematical Society
Publication date: 1/13/2009
Binding: Hardcover
Pages: 590
Size: 6.25" wide x 9.00" long x 1.00" tall
Weight: 2.750
Language: English

Preliminaries
Tools for analysis
Convergent sequences
Continuous functions Differentiation
Elementary functions as solutions of differential equations
Integration: Two fundamental theorems
Integration: Further topics
Approximation by Taylor polynomials
Sequences and series of functions
The Euclidean space ${\mathbb R}^n$
Continuity, compactness, and connectedness Metric spaces
Differentiating functions of several variables
Local approximation of real-valued functions
Approximating nonlinear mappings by linear mappings
Images and inverses: The inverse function theorem
The implicit function theorem and its applications
Integrating functions of several variables
Iterated integration and changes of variables
Line and surface integrals
Consequences of the field and positivity axioms
Linear algebra
Index
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