proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

Question

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\partial x}$$.I saw this relations frequently used in the derivation of different formula for multiple gradient,curl etc. which is fundamental for many other important theorems.My doubt is that is it true for all function?.What is the original proof for this relation.My thought is that it is not true for any function.there may be some function which does not satisfy this relation.If there is such function why we uses this relation in many physical problem.any link or answer would be appreciated.

Answer 1

The Schwarz theorem is valid only if both derivatives are continuous.

So if $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}$ is continuous and $\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$ is continuous, then they're equal.

This isn't really helpful in pratice, but one can also shows that if $\frac{\partial f(x,y)}{\partial x}$, $\frac{\partial f(x,y)}{\partial y}$ and $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}$ are continuous, then also $\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$ is continuous, and then the Schwarz theorem applies.