Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics)

✍ Scribed by James J. Callahan

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 542
• Edition: 2010
• Category: Library

✦ Synopsis

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.  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.  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.  This 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.  The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.

✦ Table of Contents

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