Advanced / In-Depth Calculus Book for Self-Edification

Question

I am a pre-engineering student currently taking a Single Variable Calculus course at a community college.

I recognize that my future success (or not so much) as an engineer will be based, in large part, on my capabilities with and understanding of Calculus. Therefore, I really, really want to master it like I've never mastered any subject before.

I'm doing well in my class, and my instructor is great, but I am under the impression that this course and it's textbook (Calculus, Early Transcendentals by Stewart) do not delve quite as deeply into Calculus as I would like. Also, the textbook frequently introduces new techniques and concepts with little to no explanation.

(Incidentally, I'm a self-taught software developer, so I am adept at learning new topics on my own. Learning Mathematics is, IMHO, quite similar to learning a new programming language.)

So I'm hoping to find some really excellent Calculus textbooks that will give me deep insight into the topics of differentiation and integration (and any other topics my course may be missing).

I've used Google and my school's library to search extensively, and I've found no shortage of Calculus textbooks. My problem is that, since I'm just now learning the basics, I have no way to know just how in-depth an advanced or in-depth book should go, or what important information my current textbook may be missing.

I own a copy of The Calculus Lifesaver, by Adrian Banner, which is absolutely outstanding. If anyone reading this happens to be struggling with Calculus, this is the book to turn to.

I also have been taking advantage of the Calculus courses in MIT's OpenCourseware. Calculus Revisited, with Herbert Gross, has been very helpful. His way of explaining the concepts just really "clicks" with me.

So, with that said, I'm just hoping the experts in the community here can recommend some great resources (e.g. books, free online courses, or other media) to help me optimize my knowledge of Calculus. Thanks in advance!

Answer 1

I recommend Advanced Calculus by G. B. Folland. It is rigorous and still elegant, especially when it comes to vector calculus.

Answer 2

From what I gather the holy grail of calculus books seems to be Calculus by Micheal Spivak.

Answer 3

When I was a freshman in college, we used Calculus by Tom Apostol. It took me many years to appreciate its greatness. However, if you'd like to learn the subject in depth, this book is really worth sinking your teeth into.

Answer 4

If you want CALCULUS to be fun, you can also read these two along with Spivak

• Calculus for Dummies
• Calculus II for Dummies

Don't just go on there names, these are not only for Dummies. Here is what author has to say-

I personally like those books which talks to you while you are reading them. It does and it is humorous too!

Answer 5

I recommend the two volumes of Calculus by T. Apostol, they are a classic.

Answer 6

I would suggest using Ross' Elementary Analysis: The Theory of Calculus. After reading that, you should peruse Rudin's Principles of Mathematical Analysis since it is the indisputable bible of elementary analysis.

Answer 7

I strongly recommend Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard. (Get it off their website, it is far cheaper than copies off Amazon)

Given that you are a student of engineering, it is important to feel comfortable with the coexistence of linear algebra with calculus, which is something many calculus textbooks miss (because linear algebra typically isn't a prerequisite for calculus)!

Furthermore, this book will cover vector calculus in the more general setting of manifolds, giving you the deep dive I think you're looking for. There is a thorough, deep chapter on single and multivariable integration, with a short treatise to Lebesgue integration. This book discusses differential forms in depth, a topic many calculus textbooks overlook (it also discusses their applications).

I'll leave you with the author's agenda lined out in the preface:

First, we believe that at this level linear algebra should be a more convenient setting and language for multivariate calculus than a subject in its own right. The guiding principle of this unified approach is that locally, a nonlinear function behaves like its derivative.

Second, we emphasize computationally effective algorithms, and we prove theorems by showing that these algorithms work.

Third, we use differential forms to generalize the fundamental theorem of calculus to higher dimensions.