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TD
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mathwonk said:TD: I do like the approach via inverting integrals, as when generalized, it leads to the beautiful elliptic functions as inverses of the integral of 1/sqrt(1-x^4), and which are so important in algebraic geometry and number theory, e.g. in the proof of fermat's last theorem.
this approach aslso explains the trick of "separating variables" in solving d.e.'s.
i.e. by the inverse function theorem, if f' = P(x), then g'(x) = P(g(x)), where g is the inverse of f.
this is the whole basis for the so called separable variables technique, but i have never seen it so simply explained in any book. i noticed it myself this fall while teaching integral calculus and discussing exactly the "inverse of integrals" ideas we have been discussing.
We didn't really went any further than just defining the functions I described, as examples. We did see the inverse function theorem in the chapter before that, but we didn't prove it (it was said to be rather 'advanced' at that point). We used it though to prove the implicit function theorem (which was asked on the exam and I couldn't do it back then
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