# Geometric interpretation of mixed partial derivatives?

I’m looking for a geometric interpretation of this theorem:

My book doesn’t give any kind of explanation of it. Again, I’m not looking for a proof – I’m looking for a geometric interpretation.

Thanks.

Inspired by Ted Shifrin’s comment, here’s an attempt at an intuitive viewpoint. I’m not sure how much this counts as a “geometric interpretation”.

Consider a tiny square $ABCD$ of side length $h$, with $AB$ along the $x$-axis and $AD$ along the $y$-axis.

D---C
|   | h
A---B
h


Then $f_x(A)$ is approximately $\frac1h\big(f(B)-f(A)\big)$, and $f_x(D)$ is approximately $\frac1h\big(f(C)-f(D)\big)$. So, assuming by $f_{xy}$ we mean $\frac\partial{\partial y}\frac\partial{\partial x}f$, we have

Similarly,

But those two things are the same: they both correspond to the “stencil”