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So, we know from thermodynamics that (dy/dx)(dx/dz)(dz/dy), where the d's represent partial derivatives, is equal to -1, provided that z is a function of x and y. There are several proofs of that. My questions are:

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)?

Wikipedia (https://en.wikipedia.org/wiki/Triple_product_rule#Derivation) provides with a derivation for the n=3 case. In the first proof, they set dz=0 in the third step. Doesn't that make the proof lose generality? What do you think of both proofs Wikipedia provides, from a mathematical point of view?

Thanks in advance for your help!

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Both of the derivations on Wikipedia proofs are quite incomprehensible to me. The only way I can understand this result is as I have explained in another answer.

If you look at the proof there, you will hopefully easily be able to generalize it to more variables, and see that indeed you get $(-1)^n$.

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