Generalization of the triple product rule

Asked Sep 27 '16 at 20:59

Active Sep 28 '16 at 09:59

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$$\left(\frac{\partial y}{\partial x} \right)\left(\frac{\partial x}{\partial z} \right) \left(\frac{\partial z}{\partial y} \right) = -1$$

provided that $z$ is a function of $x$ and $y$.

My question is: Is there a generalization of this identity for $n$ consecutive partial derivatives in a circular permutation, and is that the superintuitive (-1)n ? If so, can someone provide the link of a proof (or write it down, maybe)?

Plus, why haven't I found anything on this topic online? Isn't it possible to find such expressions in thermodynamics (fundamental relations for instance)? How do we "attack" these expressions?

edited Sep 28 '16 at 09:59

Qmechanic

12,298

asked Sep 27 '16 at 20:59

J.O'Connor

6

How exactly do we know that from thermodynamics? – Sep 27 '16 at 21:01
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Here is the wiki page for the identity you are after. This is a pure maths question and should probably be moved to math SE. – By Symmetry Sep 27 '16 at 21:47
Crossposted to http://math.stackexchange.com/q/1944126/11127 – Qmechanic Sep 28 '16 at 10:10

Generalization of the triple product rule

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