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I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is,

$$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$

where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that $$x_i = \frac{\partial G}{\partial s_i}$$

However, doesn't this come from the definition itself? I mean for one variable, the transformation is, $$G(s) = s x(s) - F(x(s))$$ such that $x(s) = dG/ds$. So how do I prove the above statement?

Amy Tratner
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  • If you start from $s_i = \frac{\partial F}{\partial x_i}$, how would you go about getting an expression for $F(x_1,...,x_n)$? From there, how would you introduce a function $G(s_1,...,s_n)$? – Endulum Jan 23 '15 at 03:17
  • Essentially a duplicate of http://physics.stackexchange.com/q/105912/2451 and links therein. – Qmechanic Jan 26 '15 at 14:14

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