Legendre transformation

Let df(x,y)=u(x,y)dx+v(x,y)dy. Substituting g(u,y)=f(x,y)-xu(x,y) gives dg=vdy-xdu. Legendre Transformations are used in thermodynamics, and to derive Hamiltonian mechanics from Lagrangian mechanics. A Legendre transformation is its own inverse, and is related to integration by parts.

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