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I need advice on my studies of mathematics... I'm really depressed because it's impossible for me to understand many important parts of books such as Tenenbaum & Pollard "Ordinary Differential Equations" or Kreyszig's "Differential Geometry", even after having got an A at a rigorous course in calculus (construction of the reals, limits with epsilon-delta arguments, proofs of almost all theorems presented...).

The reason is that these books on DE and DG use thinks like multiply the two sides of the equation by dx, or integrate dt, consider an infinitesimal displacement, etc, to arrive at conclusions... I really can't understand this reasonings... And I'm now looking at books on mechanics (for engineers) and it's even worse, because they talk about "virtual work", and other impossible-to-understand (for me) things...

What should I do? Relearn calculus from some textbook that teach these things or maybe search other books for learning differential equations, mechanics, and differential geometry? I'm feeling really dumb.

Thanks in advance.

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closed as off-topic by user147263, JMP, user99914, Claude Leibovici, Harish Chandra Rajpoot Nov 11 '15 at 7:04

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I think that their difficulties lie more in the opposite direction. They're used to a certain sort of rigor along with clear definitions of objects and their properties, and the arguments given in those sorts of books (at least for DE) can be somewhat hand-wavy with those things because they're trying to approach the subject from an intuitive standpoint. From here, I think you have two options:

The first is to go through a rudimentary, non-rigorous calculus text and try and match up your rigorous definitions/proofs to their more hand-wavy ones. That might be able to help you make the connection between what you know rigorously, and the kind of 'calculus intuition' that the DE and DG books are expecting of you.

The second is to find the rigorous treatment of those subjects in other texts, which will make use of what you already know and the way in which you know it, but may not always be the best for actual problem-solving.

Personally, I'd recommend doing a little bit of both. Intuition can go a long way, even in pure math, so it's important that you gain an intuitive feel for what you're doing while simultaneously understanding in a more rigorous manner why those hand-wavy arguments work the way that they do and what they're actually saying. Hope that helps!

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