Advanced Calculus: A Differential Forms Approach

✍ Scribed by Harold M. Edwards

• Publisher: Springer Science & Business Media
• Year: 2013
• Tongue: English
• Leaves: 528
• Series: Modern Birkhäuser Classics
• Category: Library

✦ Synopsis

In a book written for mathematicians, teachers of mathematics, and highly motivated students, Harold Edwards has taken a bold and unusual approach to the presentation of advanced calculus. He begins with a lucid discussion of differential forms and quickly moves to the fundamental theorems of calculus and Stokes’ theorem. The result is genuine mathematics, both in spirit and content, and an exciting choice for an honors or graduate course or indeed for any mathematician in need of a refreshingly informal and flexible reintroduction to the subject. For all these potential readers, the author has made the approach work in the best tradition of creative mathematics. This affordable softcover reprint of the 1994 edition presents the diverse set of topics from which advanced calculus courses are created in beautiful unifying generalization. The author emphasizes the use of differential forms in linear algebra, implicit differentiation in higher dimensions using the calculus of differential forms, and the method of Lagrange multipliers in a general but easy-to-use formulation. There are copious exercises to help guide the reader in testing understanding. The chapters can be read in almost any order, including beginning with the final chapter that contains some of the more traditional topics of advanced calculus courses. In addition, it is ideal for a course on vector analysis from the differential forms point of view. The professional mathematician will find here a delightful example of mathematical literature; the student fortunate enough to have gone through this book will have a firm grasp of the nature of modern mathematics and a solid framework to continue to more advanced studies. The most important feature...is that it is fun—it is fun to read the exercises, it is fun to read the comments printed in the margins, it is fun simply to pick a random spot in the book and begin reading. This is the way mathematics should be presented, with an excitement and liveliness that show why we are interested in the subject. —The American Mathematical Monthly (First Review) An inviting, unusual, high-level introduction to vector calculus, based solidly on differential forms. Superb exposition: informal but sophisticated, down-to-earth but general, geometrically rigorous, entertaining but serious. Remarkable diverse applications, physical and mathematical. —The American Mathematical Monthly (1994) Based on the Second Edition

✦ Table of Contents

Preface to the 1994 Edition
Introduction
Preface
Contents
Synopsis
chapter 1 constant forms
1.1 One-Forms
1.2 Two-Forms
1.3 The Evaluation of Two-Forms Pullbacks
Examples
1.4 Three-Forms
1.5 Summary
chapter 2 integrals
2.1 Non-Constant Forms
2.2 Integration
2.3 Definition of Certain Simple Integrals. Convergence and the Cauchy Criterion
2.4 Integrals and Pullbacks
2.5 Independence of Parameter
2.6 Summary. Basic Properties of Integrals
chapter 3 integration and differentiation
3.1 The Fundamental Theorem of Calculus
Proof of I
Proof of II
3.2 The Fundamental Theorem in Two Dimensions
3.3 The Fundamental Theorem in Three Dimensions
3.4 Summary. Stokes' Theorem
Terminology and notation
chapter 4 linear algebra
4.1 Introduction
4.2 Constant k-Forms on n-Space
Definitions
Definition
Theorem
Proof
4.3 Matrix Notation. Jacobians
1-forms
2-forms
3-forms
4.4 The Implicit Function Theorem for Affine Maps
Implicit Function Theorem
Proof
Solution in Closed Form
4.5 Abstract Vector Spaces
Theorem
Corollary
Proof
Definition
Glossary
4.6 Summary. Affine Manifolds
chapter 5 differential calculus
5.1 The Implicit Function Theoremfor Differentiable Maps
Implicit Function Theorem
5.2 k-Forms on n-Space. Differentiable Maps
Chain Rule
Implicit Differentiation
Examples
5.3 Proofs
Proof of the Chain Rule
Implicit Differentiation
Implicit Function Theorem
Elimination Theorem
5.4 Application: Lagrange Multipliers
Example
Example
Method of Lagrange Multipliers
Proof
Example
Example
Example
Example
The Holder Inequality
Example
Example
Example
Example
Example
Example
Example
5.5 Summary. Differentiable Manifolds
Elimination Theorem
chapter 6 integral calculus
6.1 Summary
6.2 k-Dimensional Volume
Definition
Theorem
Proof
6.3 Independence of Parameterand the Definition of ∫sw
Example
Theorem
Proof
Lemma
Independence of Parameter
lemma
Proof
6.4 Manifolds-with-Boundary and Stokes' Theorem
Definition
Theorem
Proof
Stokes' Theorem
Proof
6.5 General Properties of Integrals
Definition
Theorem
Stokes' Theorem
Proof
Proofs
6.6 Integrals as Functions of S
Definition
chapter 7 practical methods of solution
7.1 Successive Approximation
Elimination Theorem
Proof
7.2 Solution of Linear Equations
7.3 Newton's Method
7.4 Solution of Ordinary Differential Equations
Theorem
Proof
7.5 Three Global Problems
Problem 1
Problem 2
Problem 3
chapter 8 applications
8.1 Vector Calculus
Line Integrals
Surface Integrals
Volume Integrals
8.2 Elementary Differential Equations
Example
Example
Example
Example
Example
Example
Example
Example
8.3 Harmonic Functions and Conformal Coordinates
Theorem
Proof
8.4 Functions of a Complex Variable
Theorem
Corollary
Proof of Corollary
Proof of Theorem
The Binomial Series
The Fundamental Theorem of Algebra
The Implicit Function Theorem for Analytic Functions
Harmonic Functions are Analytic
Poisson Integral Formula
8.5 Integrability Conditions
Theorem
Proof
8.6 Introduction to Homology Theory
Theorem
Theorem
Theorem
Poincare's Lemma
Proof
8.7 Flows
Theorem
Theorem
8.8 Applications to Mathematical Physics
The Heat Equation
Potential Theory
Maxwell's Equations
Special Relativity
Mass and Energy
chapter 9 further study of limits
9.1 The Real Number System
Natural Numbers
Rational Numbers
Real Numbers
Theorem
Proof
9.2 Real Functions of Real Variables
9.3 Uniform Continuity and Differentiability
Theorem
Definition
9.4 Compactness
Theorem 1
Theorem 2
Bolzano-Weierstrass Theorem
Heine-Borel Theorem
Proof of the Bolzano-Weierstrass Theorem
Proof of the Heine-Borel Theorem
Proof of Theorem 2
Proof of Theorem 1
Theorem 3
Proof
9.5 Other Types of Limits
Infinite Series
Infinite Products
Improper lntegrals
Examples and Applications
9.6 Interchange of Limits
9.7 Lebesgue Integration
Theorem
Examples and Applications
Proof
Definition
Theorem
Examples and Applications
lebesgue Dominated Convergence Theorem
Theorem
Lemma
Proof
9.8 Banach Spaces
Implicit Function Theorem
appendix 1 the Cauchy Criterion
appendix 2 the Leibniz notation
appendix 3 on the foundations of mathematics
appendix 4 constructive mathematics
appendix 5 the parable of the logician and the carpenter
answers to exercises
index