Advanced Calculus: A Geometric View / Edition 1

Advanced Calculus: A Geometric View / Edition 1

by James J. Callahan
     
 

ISBN-10: 1441973311

ISBN-13: 9781441973313

Pub. Date: 09/17/2010

Publisher: Springer New York

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones—the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes—the text treats other important topics in differential

Overview

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones—the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes—the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. This invites geometric visualization; the book incorporates modern computational tools to give visualization real power.

Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps, such as the role of the derivative as the local linear approximation to a map and its role in the change of variables formula for multiple integrals. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books.

Advanced Calculus: A Geometric View is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses—including topics in geometry. It avoids duplicating the material of real analysis. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.

Product Details

ISBN-13:
9781441973313
Publisher:
Springer New York
Publication date:
09/17/2010
Series:
Undergraduate Texts in Mathematics Series
Edition description:
2010
Pages:
526
Sales rank:
747,802
Product dimensions:
7.10(w) x 10.00(h) x 1.40(d)

Table of Contents

1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morse’s lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Green’s Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokes’ Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokes’ Theorem.- 11.4 Closed and Exact Forms.- Exercises

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