Welcome to mecmath.net

This site is the homepage for the free book Vector Calculus, by Michael Corral (Schoolcraft College). If you are looking for the Elementary Calculus homepage, go here. If you are looking for the Trigonometry homepage, go here.

You can download the latest version (2013-05-21) of Vector Calculus here: calc3book.pdf

See the News section below to read about the changes in this version.

Note: The above file gets updated only with corrections (of typos, errors, etc). No new material gets added, and no material gets removed.

The book is a PDF file, which requires a PDF viewer such as the freely available Adobe Acrobat Reader or noncommercial software (e.g. xpdf, Evince) to view it. The book is distributed under the terms of the GNU Free Documentation License, Version 1.2.

The PDF file above includes color graphics. You can buy a printed, bound, paperback version of the book with grayscale graphics for $11 plus shipping at the book's project page on Lulu.com: http://www.lulu.com/mecmath.

The price covers just the cost of printing (Lulu.com recently increased their base price, which is why the price is now $11 instead of the original $8.93). The grayscale version is also available as a free download at that site.

For those who want to view and/or compile the book's source files, the LaTeX source code is available here: calc3book-1.0-src.tar.gz

You will need a relatively recent LaTeX installation to compile the source code. A build script (calc3book.sh) for Linux/UNIX systems is included for building the PDF file. See the included README file for more instructions. A brief 10-page mini-tutorial on LaTeX that I wrote is available here.

Java examples: The Java programs and source code for Newton's algorithm (Ch. 2) and the Monte Carlo method (Ch. 3) can be downloaded here: calc3book_java.zip

These are the programs listed in Section 2.6 and Section 3.4 of the book. The source files can be modified easily to solve some of the exercises in those sections.

(2013-05-21) Matlab/Octave equivalents of the above Java programs can be downloaded here: ParallelizationArea.zip

These were kindly provided by Prof. Benson Muite (University of Michigan).

(2012-02-13) Sage equivalents of the above Java programs can be downloaded here: calc3book_sage.zip

Book description: This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

The book also includes discussion of numerical methods: Newton's method for optimization, and the Monte Carlo method for evaluating multiple integrals. There is a section dealing with applications to probability. Appendices include a proof of the right-hand rule for the cross product, and a short tutorial on using Gnuplot for graphing functions of 2 variables.

There are 420 exercises in the book. Answers to selected exercises are included.

TABLE OF CONTENTS

Vectors in Euclidean Space Introduction

Vector Algebra

Dot Product

Cross Product

Lines and Planes

Surfaces

Curvilinear Coordinates

Vector-Valued Functions

Arc Length Functions of Several Variables Functions of Two or Three Variables

Partial Derivatives

Tangent Plane to a Surface

Directional Derivatives and the Gradient

Maxima and Minima

Unconstrained Optimization: Numerical Methods

Constrained Optimization: Lagrange Multipliers Multiple Integrals Double Integrals

Double Integrals Over a General Region

Triple Integrals

Numerical Approximation of Multiple Integrals

Change of Variables in Multiple Integrals

Application: Center of Mass

Application: Probability and Expected Value Line and Surface Integrals Line Integrals

Properties of Line Integrals

Green's Theorem

Surface Integrals and the Divergence Theorem

Stokes' Theorem

Gradient, Divergence, Curl and Laplacian

Bibliography

Appendix A: Answers and Hints to Selected Exercises

Appendix B: Proof of the Right-Hand Rule for the Cross Product

Appendix C: 3D Graphing with Gnuplot

GNU Free Documentation License

History

Index

News

(2013-05-21) I finally(!) got around to uploading the Matlab/Octave versions of the programs in the book, which Prof. Benson Muite (Univeristy of Michigan) kindly sent me over a year ago. I apologize for the delay; my only excuse is that my schedule became incredibly hectic over the last year. Now that things have settled down again, I should have some time to start working on a French translation of Vector Calculus (as well as finish Elementary Calculus). There have been many offers from people around the world to translate my books into other languages. For example, Prof. Koichiro Yamashita will post his Japanese translation at http://kymst.net after he finishes it.

The latest version of Vector Calculus contains a correction of a typo in one of the plots (Fig. 1.8.3 on p.54), which Prof. Yamashita found.

(2012-02-13) I ported the Java code examples in Sections 2.6 and 3.4 to Sage, a powerful and free open-source mathematics software system that is gaining in popularity. The Sage code examples are in the calc3book_sage.zip file, and can be run either on the command-line or as worksheets in a Sage notebook. See the included README file for more details.

The reason for doing this is because I received a request a few years ago to rewrite those code examples for Sage. I wasn't as familiar with Sage as I am now, so I finally got around to doing it. In general, I will be using Sage more, and in particular it will be used extensively for the code examples in my upcoming Elementary Calculus book.

(2011-06-29) The latest version of the book is out. The content of the book is basically the same as before. The big change was in switching the math font from txfonts to Fourier-GUTenberg. This was done to make the fonts more consistent. In particular, the fouriernc package makes use of the New Century Schoolbook normal text fonts for numbers and letters in math mode. This way there is no longer the incongruity of having txfonts' Times Roman-like numbers and letters in math mode versus New Century Schoolbook's different-looking numbers and letters in the main text. This change required a bit of space adjustment throughout the text, since some of the symbols in the Fourier-GUTenberg fonts are slightly smaller than those in txfonts. The sans serif font was also changed, from Avant Garde to Helvetica.

Another change was cleaning up the graphics, which also had a mish-mash of inconsistent fonts and other issues (in particular the graphics created with MetaPost and Gnuplot). The Gnuplot graphics were slightly improved over some of the default settings which I had used originally.

These changes in the appearance make the book look better overall, in my opinion, and were long overdue. It also brings the book in line with the general look and feel of my Trigonometry book and my forthcoming Elementary Calculus book (the prequel to this book).

As far as the content in the main text itself, the only changes are:

I added a tiny clarification on the relationship between determinants and volumes of parallelepipeds, right before Theorem 1.17 in Section 1.4. In particular I give the conditions for when the determinant gives the positive volume or the negative volume.

In Section 1.7 I added a footnote about the left-handedness of the usual definition of the spherical coordinate system used by mathematicians. I did this because physics students may get confused when they see the definitions of θ and φ switched in their physics classes.

Improved the code listings to use a monospaced font (Bitstream Vera Sans Mono). I still do not know what possessed me to use a proportional font originally.

Updated the URL for downloading Java (since Oracle bought Sun Microsystems).

Four corrections in the answers in Appendix A: 1.5 #1, 1.9 #3, 4.1 #11, 4.5 #4 (thanks to P. Taskas and G. Strzalkowski).

Updated instructions for using Gnuplot in Appendix C (in the Windows version some of the defaults and procedures changed slightly).

Update (2011-06-30): The printed bound version of the book on the Lulu.com site has also been updated with the latest changes.

(2011-04-17) I've written up a very short (10-page) mini-tutorial on using the LaTeX typesetting system. You can download it here: latex-tutorial.pdf

The source code for the tutorial is available here: latex_tutorial.zip

The tutorial was originally created for students in a class I'm teaching this semester, and I've expanded it a bit since then. I hope others find it useful.

(2010-06-06) Typos in the proof of Theorem 1.20(f) on p.53 have been corrected (thanks to F. Dockhorn for finding those). A newer version of the TikZ/PGF graphics package broke the diagrams on pp.60-61, so the code for those diagrams has been updated. Also, I am still working on the prequel - Elementary Calculus - which (barring a miraculous increase in my productivity) will likely not be ready until sometime next year.

(2009-09-13) The author's new book, Trigonometry, is now available. The homepage is located here: mecmath.net/trig/

(2009-07-22) The prequel to this book, which will be titled Elementary Calculus, is in preparation. It will cover calculus of a single variable. The aim is to have it available by the end of this year or early next year. Another book, on trigonometry, is almost finished and should be available here by the end of August 2009. Both books will be free and released under the GNU Free Documentation License, complete with the LaTeX source code.

(2009-07-22) In Appendix A, the answer to Exercise 5 in Section 2.3 was corrected. Thanks to E. Cavazos for pointing out the error. This is the only change in the new version (2009-07-22) from the previous version (2009-03-29).

(2009-07-10) Prof. Marshall Hampton of the University of Minnesota, Duluth has kindly posted some notes on compiling the LaTeX source code for the book under OS X, which you can read here.

Last updated: May 21, 2013