Modern Multidimensional Calculus

by Marshall Evans Munroe

Overview

A second-year calculus text, this volume is devoted primarily to topics in multidimensional analysis. Concepts and methods are emphasized, and rigorous proofs are sometimes replaced by relevant discussion and explanation. Because of the author's conviction that the differential provides a most elegant and useful tool, especially in a multidimensional setting, the notion of the differential is used extensively and matrix methods are stressed in the study of linear transformations.
The first three chapters offer introductory material on functions and variables, differentials, and vectors in the plane. Succeeding chapters examine topics in linear algebra, partial derivatives, and applications as well as topics in vector differential calculus. The final chapters explore multiple integrals in addition to line and surface integrals. Exercises appear throughout the text, and answers are provided, making the book ideal for self-study.

Product Details

ISBN-13:	9780486840062
Publisher:	Dover Publications
Publication date:	05/15/2019
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	400
File size:	31 MB
Note:	This product may take a few minutes to download.

About the Author

M. E. Munroe was a Professor of Mathematics at the University of New Hampshire.

Read an Excerpt

PREFACE

This is a second-year calculus text devoted primarily to topics in multidimensional analysis. These topics have been developed in accordance with three major guidelines:

1. A first course in multidimensional calculus becomes far too heavy for most students if it is burdened down with rigorous proof s of all pertinent results. Therefore, the emphasis here is on concepts and methods. Where proofs are required but omitted, this is clearly pointed out.

2. When properly used the differential becomes an extremely elegant tool in the development of calculus, particularly in a multidimensional setting. Thus, the notion of differential is developed in its most potent, modern form and then used as extensively as possible.

3. A major contribution of differential calculus is to reduce nonlinear transformations to linear ones, thus making linear algebra an important part of the subject. Matrix methods are the most efficient in the study of linear transformations and hence are used extensively.

Certain innovations are necessary to accomplish these ends. This is particularly true in laying a proper foundation for an intellectually respectable and yet maximally useful theory of the differential.

The first step in this program is to insist on a careful distinction between a point in the plane and the ordered pair of real numbers consisting of its rectangular coordinates. Indeed, coordinate variables, such as x and y, must be recognized as mappings. For example, x is a symbol for the mapping that carries each point p into its abscissa x(p). Mappings of this type are the major concern here; so they must be clearly recognized.

Another type of mapping that plays a completely different role is a mapping that carries n-tuples of numbers into numbers. Suppose that f maps number pairs into number s and x, y and z, map points of the plane into numbers. Then, the equation z = f(x, y) states that the mapping z is the same as the composite mapping consisting of the ordered pair of mappings x, y followed by the mapping f. This is a typical relation in multidimensional calculus and clearly x, y, and z belong in one category here with f in another. To distinguish between these two types of mappings, the following terminology is introduced. Point-to-number mappings such as x , y, and z are called variables. This is inspired by the phrase "coordinate variable," although the word variable assumes a more general meaning here. Connecting mappings, such as f above, are called functions.

It is possible to develop a reasonably complete calculus of functions in which variables do not appear at all. However, the variables connect calculus with geometry, physics, etc.; so a calculus of functions only is strictly an "ivory tower" discipline. More germane to the applications is a calculus of variables in which functions play an essential implicit role but seldom need to be mentioned explicitly. This latter point of view is the one adopted here.

Differentials, then, need to be defined for variables. Chevalley has given a definition that accomplishes exactly this, and a suitable specialization of his development of the subject is employed here. It is principally in this respect that this calculus book differs from others of recent vintage. It has been recognized for some time now that the "dx =Δx ···" approach leads to nothing but double talk, but the usual procedure in making the differential respectable is to employ the Frechet definition. Though it involves a more sophisticated procedure, the Chevalley approach is used here because it defines directly the differential of a variable, whereas the Frechet definition yields the differential of a function.

The first three chapters contain more or less introductory material. Chapter 1 establishes the hierarchy of mappings to be studied; that is, variables connected by functions. Chapter 2 develops the one-dimensional case of the differential. The Chevalley definition of a differential is an algebraic maneuver in which the dimension of the domain of the variables makes a very minor difference. The purpose of Chapter 2 is to introduce this algebraic procedure in a setting in which each step can be easily interpreted in terms of a concrete geometric picture. The algebra is then repeated in Chapter 5 for the n-dimensional case where pictures are not so practical. Chapter 3 introduces vectors in the plane and gives some illustrations of procedures that make full use of an effectively defined differential. Topics covered in these three chapters are normally included in an introductory calculus course. The revised treatment of selected topics given here should serve to give a picture of one-dimensional calculus consistent with the development to follow for the multidimensional case.

The serious study of multidimensional calculus begins in Chapter 4 with the development of the matrix algebra that will be needed. Chapter 5 introduces the differential. Chapters 6 and 7 present topics in differential calculus, notably max-min problems, transformations and chain rules, and vector derivative operators. Chapter 8 should be review. It treats iterated integrals essentially as they are treated in any introductory calculus book. Chapters 9 and 10 are concerned with multiple integrals. Here again the differential is used as much as possible. The exterior product of differentials is defined concretely and used in the definition of the integral. Exterior algebra is thus not merely a way of systematizing known results. Instead, it becomes an effective tool in developing such things as the substitution theorem and a generalized Stokes' theorem that specializes to the classical theorems of Green, Gauss, and Stokes.

M. E. M.

Durham, N. H. October, 1962

(Continues…)

---

Excerpted from "Modern Multidimensional Calculus"
by .
Copyright © 2019 Marshall Evans Munroe.
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---

Table of Contents

CONTENTS
Chapter 1. Function and Variables
1-1 Elementary functions
1-2 Coordinate variables
1-3 Derivative and integrals
1-4 Polynomial approximations
Chapter 2. Differentials
2-1 Differentials: geometric picture
2-2 Differentials: algebraic theory
2-3 Manifold
2-4 Nigher-order derivatives
2-5 Line integrals
2-6 Comments on basic concepts
Chapter 3. Vectors in the Plane
3-1 Vectors
3-2 Differential geometry of plane curves
3-3 Area
3-4 Plane motion
Chapter 4. Topics in Linear Algebra
4-1 Matrix algebra
4-2 Linear transformations
4-3 Determinants
4-4 Elementary operations
4-5 Vectors in three dimensions
4-6 Orthonormal bases
4-7 Rotations
4-8 Quadratic and bilinear forms
4-9 Analytic geometry: lines and planes
Chapter 5: Partial Derivatives
5-1 Functions on n-tuples
5-2 Partial derivatives of functions
5-3 Partial derivatives of variables
5-4 Differentials
5-5 Geometric representations
5-6 Foundations of the calculus of variables
Chapter 6. Applications
6-1 Geometry in polar coordinates
6-2 Implicit relations, one-dimensional loci
6-3 Constrained maxima and minima, one dimension
6-4 Constrained maximan and minima, several dimensions
6-5 Chain rules
6-6 Inversion
6-7 Implicit relation, multidimensional loci
Chapter 7. Vector Differential Calculus
7-1 Introduction
7-2 Gradients
7-3 Divergence and curl
7-4 Coordinate-free definitions
7-5 Curvilinear coordinates
Chapter 8. Iterated Integrals
8-1 Twofold iterated integrals
8-2 Applications
8-3 Quadric surfaces
8-4 Threefold iterated integrals
Chapter 9. Multiple Integrals
9-1 Oriented manifold
9-2 Exterior products
9-3 Multiple and iterated integrals
9-4 Change of variable
9-5 Mass
9-6 Probability
9-7 Moments
Chapter 10. Line and Surface Integrals
10-1 Line integrals, recapitulation
10-2 Surface integrals
10-3 Surface area
10-4 Stokes type theorems
10-5 Vector integral calculus
10-6 Integrals independent of the path
ANSWERS
INDEX