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I am asking this question as a question similar to what has been asked previously for other topics as well as math in general. But I'd like to ask for text references specifically in the domain of multivariable/vector calculus.

So my question shares a similar spirit to other questions such as

Learning through guided discovery https://mathoverflow.net/questions/119621/learning-through-guided-discovery

Big list of guided discovery textbooks Big list of "guided discovery" books

Are there any books that take a 'theorems as problems' approach? https://mathoverflow.net/questions/12709/are-there-any-books-that-take-a-theorems-as-problems-approach

The discussions and responses on all those discussions are great, but they did not talk about anything for multivariable calculus.

I would greatly appreciate it if anyone could share any books they know that would fulfill some of the characteristics of a guided discovery text. Just to give a clearer sense of what kind of book I'm looking out for, the books

Excursions in Calculus http://www.amazon.com/Excursions-Calculus-Continuous-Mathematical-Expositions/dp/0883853175

and

Excursions in Classical Analysis www.maa.org/press/maa-reviews/excursions-in-classical-analysis

are great books which I've been looking for in the topic of single variable calculus, and have found them. I just hope that there exists something similar in multivariable calculus.

To start off the sharing and listing, I thought I would say that the book

Div, Curl and All That www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

to be a possible candidate, but from what I've seen from it so far, it looks like it does not have the same level of mathematical rigor as what the other two textbooks have (for the topic single variable calculus that they address).

Thank you very much!

  • I wanted to post up the links completely for all the other references, but I couldn't because my reputation isn't high enough. So pardon the incomplete links :p
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AKKA
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  • Alan Macdonald's Vector and Geometric Calculus might be the type of book you're looking for. You'll probably need to read his first book Linear and Geometric Algebra beforehand, though. But they're both pretty short and if you're already familiar with linear algebra (which you should be before starting on multivariable calculus) then you'll be able to skip some parts of LAGA. –  Dec 02 '15 at 02:02
  • Just took a look at its preface from Amazon preview; Macdonald does state in the preface that this book is a sequel to his other book on Linear and Geometric Algebra. Do you think its still ok to delve straight into Vector and Geometric Calculus without first going through his Linear Algebra book? – AKKA Dec 02 '15 at 02:06
  • Just as a note on Div, Grad, Curl and All That: it's a great book full of geometric intuition, but it's missing a lot of standard material from multivariable calculus. So while I do recommend reading it, I wouldn't use it as your only text. –  Dec 02 '15 at 02:06
  • No. You really can't read VAGC without some knowledge of Geometric (Clifford) Algebra. –  Dec 02 '15 at 02:07
  • Ok, thank you very much for your suggestions here. Responding to your edits, actually in my college here we are taught vector calculus way before linear algebra. Like linear algebra is a 400 level course while vector calculus is a top level 200 course.... – AKKA Dec 02 '15 at 02:10

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