In  this article on his ‘Azimuth’ website, John Baez considers the fact that for any three related variables $\frac{\partial u}{\partial v}{|}_w\frac{\partial v}{\partial w}{|}_u\frac{\partial w}{\partial u}{|}_v=-1$ (which has an extra minus sign compared to what one might naively expect from “cancelling differentials”).

After giving an argument for the case where $u$, $v$, and $w$ are linear functions of two other variables $x$ and $y$, he asks for “a more symmetrical, conceptual proof” and goes on to promote one put forward by Jules Jacobs based on the anticommutativity of the wedge product. But I think the wedge product argument adds unnecessary formalism and infrastructure, without really clarifying the intuitive concept that makes things work out as they do.

First, the linear argument can easily be expressed more symmetrically in a way that generalizes the identity to any number of variables as follows:

Let the relation be ${\mathrm{\Sigma }}_{i\in I}{a}_i{u}_i=c$. Then $u_i=\frac{c-{\mathrm{\Sigma }}_{j\ne i\in I}{a}_j{u}_j}{a_i}$. and so $\frac{\partial u_i}{\partial u_j}{|}_{u_k:k\ne i,j}=-\frac{a_j}{a_i}$ and the cyclic identity follows easily (with the product being $(-1{)}^n$ for the case of $n$ variables).

And of course any second year calculus student should know that the linear argument can be applied locally to the case of any smooth relationship function $f(u_i,i\in I)$, giving $\frac{\partial u_i}{\partial u_j}{|}_{f,u_k:k\ne i,j}=-\frac{\frac{\partial f}{\partial u_j}{|}_{u_k:k\ne j}}{\frac{\partial f}{\partial u_i}{|}_{u_k:k\ne i}}$

So the intuition for where the minus signs come from is just the act of “moving variables to the other side of the equation”. And if the idea of partial differentials is to have any meaning then the place to start worrying about those minus signs is not in the general cyclical identity but in the simple two variable case of implicit differentiation where $\frac{du}{dv}{|}_{f(u,v)=c}=-\frac{\frac{\partial f}{\partial v}{|}_u}{\frac{\partial f}{\partial u}{|}_v}$.

I need to think some more about this – in particular how the minus signs from implicit diff in “my” argument (or from solving linear equations in the Baez linearization) relate to those from reversing wedge products in the Jacobs argument. (But looking at Jacobs’ X-post there is a reference to an article by Peter Joot on solving equations by use of wedge product which probably makes it all clear.)

Source: The Cyclic Identity for Partial Derivatives | Azimuth