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I just graduated from an engineering school and am about to begin classes as a PhD candidate. As I review my undergraduate classes and sift through old notes, I find myself wanting for more math knowledge. While I did well in math classes, I always felt as though I was following steps rather than gaining a depth of understanding and intuition. To get to the point:

Can you recommend any books to brush up on and perhaps gain a more advanced understanding of calculus and PDEs?

As an undergraduate, I used Salas Hille and Etgen, which I felt was great at teaching rote steps to solve problems but lacked some depth.

Thanks in advance!

(Questions regarding a reasonable self-guided book have very frequently been brought up on this forum. However, I like to think this is unique and apologize if this feels "repeated.")

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If you are looking for a depth of understanding of PDEs, I would suggest Partial Differential Equations: An Introduction by Walter Strauss. It is fairly concise book that gives you a taste of the theory of PDEs (such as the Maximum Principle), while not "straying" too far into theory (if you're not interested in that). The book is full of applications to the physical world, and Strauss motivates a lot of the concepts through such applications. As I recall, there is a beautiful section on the Schrödinger Equation, which doesn't require any prerequisite knowledge of physics to read. It is very self-contained, develops sufficient theory, while displaying the applications of PDE to different fields.

As for calculus, I would recommend fingering through the two-volume calculus series by Courant and Fritz John. It has a lot of examples and they do a good job with developing intuition for the subject, which may not be the case with other books as you have stated.

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Thank you for the advice. I actually came across Courant and Fritz John via Amazon lists in my earlier searches. In conjunction with your advice, it seems like a very reasonable choice. As for PDE's, I'm studying chemistry, so you've inadvertently hit the nail on the head. Many thanks! –  TexanFromTexas Sep 12 '11 at 20:15
    
I'm glad you find this of some help! –  r.g. Sep 13 '11 at 7:44

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