Reviewing single variable limits and calculus: how to go about it?

Question

I am a new math major, and I will be starting in the third year of the program (transfer from engineering) this September. I need to get comfortable taking limits of functions, differentiating them, integrating them, etc. I used to be able to do this quite well in the past, but I must admit that my knowledge was fairly mechanical. I didn't really understand stuff deep down (even if I knew the epsilon-delta definition of a limit...).

I am tempted to use Apostol's book to review, rather than a standard calculus book (Stewart, boring!). This is because I feel like I might no longer be satisfied by the more mechanical understanding that a book like Stewart might present, having been exposed (somewhat) to books by people like Courant or Gowers. When starting to read Apostol though, after he mentions Foundations of Analysis by Landau, I can't help but wonder how interesting that book might be...

Might going with Stewart indeed be the best idea, since I am just trying to reawaken deadened muscle memory? Maybe the deeper understanding can be left for the process of continuous education, while I have to realize that my immediate goal is to get muscle memory back up for school? Or, is there a short and sweet single variable calculus document, that is still insightful, that I might benefit from during my review? Perhaps that's hoping for a silver bullet...

Some wisdom and advice from teachers would be very much appreciated. Although, I am afraid the advice might just be: "this is a stupid question, just sit down and do your work even if it bores you, princess".

Edit: I thought of a good example that illustrates my concerns. If I were to go through Stewart's book for limits, I am going really be left behind the task of memorizing the limit rules, and then I'll have to apply them to various problems. On the other hand, if I go through Apostol's book, I'll know where and why each rule exists (is it an axiom, or for instance, is something like the limit sum rule (the limit of a sum is the sum of the limits) the consequence of some deeper properties of limits...?

Answer 1

I honestly think Rudin's book, "Principle of Mathematical Analysis", is a bit too dense for your needs.My classic recommendation would be "Calculus", by Spivak.

However, I've come across a couple of great books which make the subject even more bearable:

• "A First Course in Mathematical Analysis", by Brannan;

  This book is wonderful for self-study: it's very comprehensive, has meaningful examples and is rigorous enough for your needs. It's a good prequel to a more serious course in real analysis.

• "Mathematical Analysis I", by Canuto and Tabacco.

  Another great book for self-study. To some extent it's more rigorous than Brannan's book. It has lots of examples and solved exercises. (You can have a look at some extra theorems at the book's website). There's also a sequel to this book by the same authors, which deals with multivariable calculus: "Mathematical Analysis II".

• "Real Analysis for the Undergraduate: With an Invitation to Functional Analysis", by Pons

  The book by Pons is the most rigorous of my suggestions and will be very handy if/when you take real analysis.

PS: I'm not a teacher. My suggestions are based on the fact that you're moving from engineering to math.