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Geometrically understanding calculus of inverse functions (2023)
The Legendre transformation tells you how the inverse map acts on integrals
We’ll approach this with as much geometric intuition as possible, avoiding the dry application of formulas. Instead of approaching the inverse function theorem through formulas, we’ll explore it geometrically—it’s much more intuitive and enjoyable! \[\begin{align*} \text{IFT:} \quad & g'(y) = \frac{1}{f'(g(y))} \\ \text{Legendre:} \quad & G(y) = y\times g(y) - F(g(y)) + C \end{align*}\] For more ways to understand the ideas covered in this post you can look at the article I’ve written for the Oxford Invariants journal, as well as the wiki page on the integral of inverse functions.
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