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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
atBerkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations
Engineering from the University of Michigan.
A
This text was typeset in L TEX 2ε 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.

Copyright © 2008 Michael Corral.
Permission is granted to copy, distributeand/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 issuitable 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 important for
students to see full-blown proofs - since that is how mathematics works - toomuch rigor and
emphasis on proofs can impede the flow of learning for the vast majority 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 inthis 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 Bexercises 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 computer 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 thereader can figure
out what is being done even without knowing Java. Those exercises 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 scripting language like Python, and perhaps even
easier with a functional programming language (such as Haskell or Scheme), Java was chosen due to itsubiquity, 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 graphingfunctions 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 matter, anyone can
make as many copies of this book as desired and distribute it as desired, without needing
my...