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HELP: Legendre transform


Hello.  I understand the steps involved in taking the Legendre
transform (often found in thermodynamics derivations, and not to be
confused with the Laplace transform), but I have no intuitive feeling
for what's going on.  I.e., setting aside the formal manipulations, I
don't understand what is the gist of Legendre's idea, and I don't
think I could recognize unfamiliar situations in which taking the
Legendre transform would be useful.  Does anyone know of a Mathematica
notebook that illustrates the Legendre transform and perhaps answers
some of these questions?

Thank you very much in advance for your help.

Gabriel


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