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Vector Calculus

This is the homepage for the free book Vector Calculus, by Michael Corral (Schoolcraft College).

Latest version (2022-08-15): VectorCalculus.pdf

Java code samples from the book: calc3book_java.zip
MATLAB/Octave versions: ParallelizationArea.zip (courtesy of Prof. Benson Muite (University of Michigan))
Sage versions: calc3book_sage.zip

Note: The PDF was built using TeXLive 2011 and Ghostscript 9.53 under Linux (Fedora).
LaTeX source code: calc3book-1.0-src.tar.gz

Mini-tutorial on LaTeX: latex-tutorial.pdf

The book is distributed under the terms of the GNU Free Documentation License, Version 1.2.

You can buy a printed and bound paperback version of the book with grayscale graphics for $10 plus shipping at Lulu.com here.

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.

1. Vectors in Euclidean Space
   • Introduction
   • Vector Algebra
   • Dot Product
   • Cross Product
   • Lines and Planes
   • Surfaces
   • Curvilinear Coordinates
   • Vector-Valued Functions
   • Arc Length
2. 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
3. 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
4. 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

(2022-08-15) Corrected a few references to equation numbers in Section 4.3, and replaced a theta by phi in the spherical coordinate version of the divergence in Section 4.4 (thanks to Yanfeng Li of Tianjin University).

(2022-07-22) Corrected a typo in two examples in Chapter 3 (thanks to Yanfeng Li of Tianjin University).

(2022-05-11) Corrected a few typos in Appendix B (thanks to Yanfeng Li of Tianjin University, who is translating the book to Chinese).

When making the corrections I realized that I will need to update the LaTeX code so that it compiles with TeXLive 2020 (the compilation fails miserably right now). I was able to compile the PDF using TeXLive 2011 but it required some hacks. This might be a good time to redesign the book in general.

(2021-01-05) Cleaned up the web page to make it less hideous and more consistent with the revamped page for Elementary Calculus.

(2021-01-02) After too many years of neglect I finally got around to correcting a few typos that had been floating out there for a while:

• Appendix A: The answer to Exercise 5 from Section 1.9 is now fixed.
• Section 1.1: In Example 1.3(d) R^3 now has the correct dimension.
• Section 1.7: In Example 1.33 a minus is now a plus.
• Appendix A: The margin overrun in the answers for Section 2.4 has been removed.

Problems with my TeXLive 2014 setup had caused numerous issues when trying to compile the book. I ended up going back to TeXLive 2011 to fix all that, and it worked. So now, after many requests, I have finally restored the ability to buy a printed and bound paperback version on Lulu.com. It's even a buck cheaper than before. See the link near the top of this page.

On a side note, there are many things about the book that I would change now, after the experience of writing the Trigonometry book and especially the Elementary Calculus book, both content-wise and stylistically. I haven't decided on that yet, but if I do re-write Vector Calculus then I would keep the current version available in addition to the new version. Any decision on that wouldn't be for at least another year, though.

(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.