Suggestions for a Rigorous Multivariate Calculus Book

[vcxp]

I've spent a lot of time soul searching after having some academic "failures" in the previous semesters, and what I found is that often when I'm not studying, it's not because I can't study, but it's because the book we're using is unreadable to me.

For example, after sailing through Calculus I with knowledge I gained from high school, and bombing a test or two in Calculus II due to not reading the text, I returned to Spivak's Calculus (having flirted with it a bit in high school) and had a religious experience. That read like a novel, and the problems were hard but I enjoyed solving them. Now, I haven't had time to finish the text, but I plan to. However, the forward progression of classes marches on, and now I find myself in multi-variable Calculus (the class just started today). I'd prefer to have something pleasant to read over Stewart's cookbook, and I'm thinking about purchasing Courant Volume II. Do you guys think I'd be okay to dive into this? I read part of Volume I a while back and I remember his writing being extremely clear, which is why it was my first choice.

 

[l'Hôpital]

I really like Hubbard and Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach". It taught rigorous multivariable (with differential forms and whatnot) in an elementary way. It's also self-contained. Granted, I'd skip some parts (he gets kinda crazy with the Implicit Function Theorem and bounds in chapter 2), but overall it's a fantastic book that actually gives you some high level stuff without losing you details. For example, you won't see any "kth exterior power of the cotangent bundle" as a definition for a k-form, but something more down to Earth that relates to the determinant, a function we all know and love.

I'd give it a try if you want. It's proofy but not like SUPER proofy.

 

[fluxions]

Here are some resources I've found useful:

At about the same level of Stewart:
- Schey Div, Grad, Curl, and all that (a very fun read, great for building intuition useful in, say, classical E&M; highly recommended for a novice.)

More 'rigorous' options than Stewart, but probably accessible to you if you've read Spivak:
- Apostol Calculus Vol. 2 (very dry).
- Jerry Shurman's Notes (read: book) on multivariate calculus: http://people.reed.edu/~jerry/211/vcalc.html" (this is much better than Apostol Vol. 2 IMO)

At a level significantly higher than Stewart, but worth checking out if you've read and understood Spivak
- Sternberg/Loomis Advanced Calculus http://www.math.harvard.edu/~shlomo/"
- Munkres Analysis on Manifolds
- Spivak Calculus on Manifolds

 

[vcxp]

I really like Hubbard and Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach". It taught rigorous multivariable (with differential forms and whatnot) in an elementary way. It's also self-contained. Granted, I'd skip some parts (he gets kinda crazy with the Implicit Function Theorem and bounds in chapter 2), but overall it's a fantastic book that actually gives you some high level stuff without losing you details. For example, you won't see any "kth exterior power of the cotangent bundle" as a definition for a k-form, but something more down to Earth that relates to the determinant, a function we all know and love.

I'd give it a try if you want. It's proofy but not like SUPER proofy.

I just snagged a copy for ~$12. Thanks for the suggestion!

 

[vcxp]

Here are some resources I've found useful:

At about the same level of Stewart:
- Schey Div, Grad, Curl, and all that (a very fun read, great for building intuition useful in, say, classical E&M; highly recommended for a novice.)

More 'rigorous' options than Stewart, but probably accessible to you if you've read Spivak:
- Apostol Calculus Vol. 2 (very dry).
- Jerry Shurman's Notes (read: book) on multivariate calculus: http://people.reed.edu/~jerry/211/vcalc.html" (this is much better than Apostol Vol. 2 IMO)

At a level significantly higher than Stewart, but worth checking out if you've read and understood Spivak
- Sternberg/Loomis Advanced Calculus http://www.math.harvard.edu/~shlomo/"
- Munkres Analysis on Manifolds
- Spivak Calculus on Manifolds

"Div, Grad, Curl, and All That: An Informal Text on Vector Calculus" interests me. It's cheap, so I'll probably order it and read it. In your experience, do the different versions of that text matter all that much?

Munkres/"Calculus on Manifolds" are on my list of things to attack after I finish Spivak's Calculus when I have some free time. I wish I could focus on math exclusively this semester, but I'm taking an E&M course and a Circuits class in addition to this (still looking for decent books for those q:).

I somehow missed Sternberg & Loomis when looking around. I'll have to check it out.

Shurman's notes look great.

You guys are awesome. I think this is already more than I could possibly read in a semester, so I'm satisfied. Thanks!

 

[fluxions]

"Div, Grad, Curl, and All That: An Informal Text on Vector Calculus" interests me. It's cheap, so I'll probably order it and read it. In your experience, do the different versions of that text matter all that much?

My copy of the book is the 3rd edition. In its preface, the author says,
"This new edition constitutes a fine-tuning of its predecessor. Several new problems have been added, two other problems awkwardly worded in the earlier edition have been revised, and a diagram has been corrected. The major change involves replacing the operators div, grad, and curl by the appropriate expressions using the del operator, to bring the text into conformity with modern notational practice. ..."

So it's really up to you if the above changes are worth ~$30 (i.e, the difference in price between a used 3rd/4th edition and a used 2nd edition).

 

[vcxp]

My copy of the book is the 3rd edition. In its preface, the author says,
"This new edition constitutes a fine-tuning of its predecessor. Several new problems have been added, two other problems awkwardly worded in the earlier edition have been revised, and a diagram has been corrected. The major change involves replacing the operators div, grad, and curl by the appropriate expressions using the del operator, to bring the text into conformity with modern notational practice. ..."

So it's really up to you if the above changes are worth ~$30 (i.e, the difference in price between a used 3rd/4th edition and a used 2nd edition).

Hmmm...thanks.

 

[Sankaku]

- Jerry Shurman's Notes (read: book) on multivariate calculus: http://people.reed.edu/~jerry/211/vcalc.html" (this is much better than Apostol Vol. 2 IMO)

I hadn't seen that around before. Thanks for the link! Looks very interesting...