Vector Calc-Calc 3.pdf

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Vector

Calculus

Michael Corral

Vector Calculus

Michael Corral

Schoolcraft College

About the author:

Michael Corral is an Adjunct Faculty member of the Department of Mathematics

at Schoolcraft College. He received a B.A. in Mathematics from the University

of California at Berkeley, and received an M.A. in Mathematics and an M.S. in

Industrial & Operations Engineering from the University of Michigan.

This text was typeset in LTEX 2

with the

KOMA-Script

bundle, using the GNU

Emacs text editor on a Fedora Linux system. The graphics were created using

MetaPost, PGF, and Gnuplot.

Permission is granted to copy, distribute and/or modify this document under the terms

of the GNU Free Documentation License, Version 1.2 or any later version published

by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts,

and no Back-Cover Texts. A copy of the license is included in the section entitled

“GNU Free Documentation License”.

Preface

This book covers calculus in two and three variables. It is suitable for a one-semester

course, normally known as “Vector Calculus”, “Multivariable Calculus”, or simply

“Calculus III”. The prerequisites are the standard courses in single-variable calculus

(a.k.a. Calculus I and II).

I have tried to be somewhat rigorous about proving results. But while it is impor-

tant for students to see full-blown proofs - since that is how mathematics works - too

much rigor and emphasis on proofs can impede the flow of learning for the vast ma-

jority of the audience at this level. If I were to rate the level of rigor in the book on a

scale of 1 to 10, with 1 being completely informal and 10 being completely rigorous, I

would rate it as a 5.

There are 420 exercises throughout the text, which in my experience are more than

enough for a semester course in this subject. There are exercises at the end of each

section, divided into three categories: A, B and C. The A exercises are mostly of a

routine computational nature, the B exercises are slightly more involved, and the C

exercises usually require some effort or insight to solve. A crude way of describing A,

B and C would be “Easy”, “Moderate” and “Challenging”, respectively. However, many

of the B exercises are easy and not all the C exercises are difficult.

There are a few exercises that require the student to write his or her own com-

puter program to solve some numerical approximation problems (e.g. the Monte Carlo

method for approximating multiple integrals, in Section 3.4). The code samples in the

text are in the Java programming language, hopefully with enough comments so that

the reader can figure out what is being done even without knowing Java. Those exer-

cises do not mandate the use of Java, so students are free to implement the solutions

using the language of their choice. While it would have been simple to use a script-

ing language like Python, and perhaps even easier with a functional programming

language (such as Haskell or Scheme), Java was chosen due to its ubiquity, relatively

clear syntax, and easy availability for multiple platforms.

Answers and hints to most odd-numbered and some even-numbered exercises are

provided in Appendix A. Appendix B contains a proof of the right-hand rule for the

cross product, which seems to have virtually disappeared from calculus texts over

the last few decades. Appendix C contains a brief tutorial on Gnuplot for graphing

functions of two variables.

This book is released under the GNU Free Documentation License (GFDL), which

allows others to not only copy and distribute the book but also to modify it. For more

details, see the included copy of the GFDL. So that there is no ambiguity on this

Vector Calc-Calc 3.pdf

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