Calc III Materials

Below is a collection of resources I've obtained over the years for specifically the subject of multivariate calculus, Calc III. This includes material I've made myself, resources I've taught from, and resources that I think are particularly helpful (note the last two are not always the same thing). This list is collected partly for my own notes, but anybody perusing this page is free to use them as they see fit - I've annotated the list so that it's easier to find the best source for the reader's purposes.

Resources

Online resources

Here are a couple nice online resources one could us in learning calculus. I often go to these when I need an immediate answer to a question, or a quick practice problem. Online text-based sources are often less dense than a book, and so are good places to review. The visual sources and lecture series, meanwhile, can be very good for gaining intuition on their topics. They're low on practice problems and proofs, but high on helping to visualize concepts. These items are listed perhaps in order of most useful downwards.

• First and foremost, Paul's Online Math Notes. This is a wonderful resource for practice problems, reviews, cheat sheats, notes, example exercises, and whatever else you might need. It also contains the same volume of information for the classes taken before Calc III as well, in case you need a review of any of them. There is also material on differential equations for those taking that course. As a TA I've often used it for practice problems, or to look up a minor detail that escapes me in conversation. Unfortunately it doesn't include material on linear algebra; unlike here at Georgia Tech that course is often taken after differential equations. Still, for any other pre-proofs course this is an incredible resource to know about and use.
• Another good resource is the playlists on Calc I/II and Linear Algebra by YouTuber 3blue1brown. This isn't a written source, so it doesn't have as many nice materials as Paul's notes. Instead, it's a lecture series, so it focuses on giving a vocal and visual experience which is conducive to the viewer's understanding. In this, 3blue1brown excels. He is a wonderful lecturer with great exposition, and he also gives viewers plenty of chances to try and figure things out for themselves before giving an explanation ("pause and ponder", as he says). The visuals he uses are also top-notch. His custom Python library gives him a lot of control over the visuals he uses on screen, and he uses it well. I've often said that I didn't truly understand eigenvalues and eigenvectors until I watched his video on that subject. He hasn't done a video series on Calc III yet, but the series he has does give a lot of good intuitive grounding on the other classes, and so are very good for students who are looking for a better understanding of prior material before going into Calc III.
• One other video series to consider is Khan Academy's series on Calc III. I haven't watched this myself; in this case I defer to the expertise of 3blue1brown, who recommends this series, as well as the general popularity of Khan Academy.

Books

Here is a list of books I consider to be more or less useful resources to learn Calc III from. As with most subjects, each of these books provides a comprehensive study of its subject. In the case of Calc III, many books also cover the other calculus courses, and at least one in this list will in fact cover more than that. I have organized this list roughly in reverse order of difficulty, with the small caveat that any books that don't at least cover the other calculus courses are near the end of the list.

• The three "standard" books I know of are the book by Thomas, which is used here at Georgia Tech; the book by Stewart; and the book by Larson, Ron, and Edwards. Each of these are roughly equivalent in that they cover about the same material in about the same way. All three are roughly 16 chapter books which cover all three calculus courses within their content. I view Stewart as perhaps the most informative yet dry of them, but it was also the first book I used, and its examples of interesting polar-coordinate graphs can be cited as the source which sparked my interest in math. Larson I'm most familiar with, but I don't have much to say about it. Perhaps that it's a bit easier of a read than Stewart. Thomas is the book we use at Georgia Tech, and that is very nearly all I know about it. Again, these books are all roughly equivalent. They all contain a wealth of information done informally (relative to the other books I give below) and have a ton of computational practice problems for every section. All are good places to start with Calc III.
• A slightly more advanced book of similar content is Marsden's series on calculus. This series is not usually compiled as one book, but the three volumes do still cover all three courses. This book is more advanced in its choice of topic and its selection of practice problems. Both are more theoretical, but not too much so. The problem selection is a mix of standard computation and harder proofs/word problems. To give an example on the content choice, the Calc III book describes equations for all possible conics rather than just the standard formulae, which are aligned with the standard bases. Similarly, Marsden's book is the one which I referenced in writing my notes on the arc length parameter below. In this sense, Marsden's content is a bit more involved, but not horribly so. You can still expectto learn most of the same material as the above three books from reading it.
• I will recommend Apostol's calculus series here as well. I don't know it very well, but it has been recommended to me several times by people with a heavy interest in mathematics; for that reason I suspect it to be even more advanced than Marsden. Content-wise, in two volumes Apostol covers all of calculus and most of linear algebra, as well as an introduction to differential equations. The problems are again a combination of proofs and computational problems, though it is more explicit in its request for proofs than Marsden. For an advanced student, this is likely a good choice of book.
• A very interesting book for Calc III will serve as my departure from books which provide the full content of the calculus series. This book is Clark Bray's Multivariable Calculus. Two things really stand out to me in this book. The first is its pictures, which are hand-drawn on perhaps a tablet, and so give the book a very real, personal feel. The text supports this, as it's written in an informal, friendly style. The hand-written nature of the drawings is also useful, in the sense that the book contains plenty of sample drawings to give a feel for how to make drawings yourself. The trade-off, of course, is that they are less exact; another book might be better for finding perfect drawings. The other benefit of this book is its handling of vector analysis, i.e., the final chapter of any calculus book. The book doesn't try to be advanced, but rather to provide a flavor for the extremely interesting advanced topics without bogging the reader down with all the details. Specifically, this means there are few proofs in the text. However, these are replaced with intuitive descriptions of powerful theorems in differential geometry, thinly masquerading as the standard topics in Calc III. There are some incredible resources within that chapter specifically coming from this approach, which I have used as references occasionaly due to their usefulness. I would highly recommend this book for an interesting take on Calc III.
• I list this final book for the sake of completion. I have taught from it, but I would be careful about choosing to learn the subject directly from it. The book is Spivak's Calculus on Manifolds, and it discusses multivariable calculus at nearly the highest level one could possibly discuss it. This includes full analytic treatments of both differentiation and integration, as well as the subjects of differential forms and manifolds. It is terse and proof-heavy, and the exercises are proof-based questions given as rigorously as possible. There are very few computational questions. If you are already accustomed to proofs and are looking for a challenging exposition on multivariable calculus, then I would recommend this book; however, for a student near the level of a more standard Calc III course, I would probably consider this a bit too advanced and suggest a different book on this list.

Writings

The following are materials I have made myself. Often they are supplementary materials I have written to answer questions from students in a Calc III course. They are not all necessary for a basic understanding of the subject; in the case that they are not, they provide supplementary information which I found interesting or pertinent enough to share. These are written in LaTeX, listed in reverse chronological order, and may be downloaded by clicking the title. Abstracts detailing the contents follow.

• Regarding the Arc Length Parameter, Jan. 2022. We develop the theory of parametrization and use this explore the arc length parameter of a position vector-valued function . We consider the relationship between this and the position vector, and describe a way in which the arc length parameter can be used in calculations.