When is $\frac{\partial^2 F(x,y)}{\partial\space x \partial\space y}$ not equal to zero?

Question

I know that if $$F(x,y)=f(x)+g(x)$$, then $$\frac{\partial^2 F}{\partial x \partial y} = 0$$ but when is $\frac{\partial^2 F}{\partial x \partial y} \not= 0$

Is it when $$F(x,y) = f(x)g(x)\space \text{or} \space f(x)^{g(x)}\space \text{or} \space f\bigl(g(x)\bigr)$$ or what?

Answer 1

We have here a simple partial differential equation. Since $$\frac{\partial^2 F}{\partial x\partial y}=0$$ then $$\frac{\partial F}{\partial x}=\varphi(x)$$ for some differentiable function $\varphi$ ($F$ is constant wrt. $y$). Hence $$F(x,y)=\Phi(x)+\Psi(y)$$ for some functions $\Phi,\Psi$ regular enough. So, $$\frac{\partial^2 F}{\partial x\partial y}\ne 0$$ whenever $F$ has not the above form.

Answer 2

In all the alternatives you proposed, there is no dependence uon the $y$ variable, so as soon as you differentiate with respect to it you’ll get $0$. Just make either $f$ or $g$ or both depend on $y$.

Answer 3

This holds for most functions of two variables (except in the special case that you give).

Try with $f(x,y)=xy$.