So I know how to prove this theorem via limits or whatever and I'm okay with that.
What I'm not okay with is the interpretation. I just can't visualise how this is true in 3d space, any ideas? How do you guys interpret this theorem?
My guess is that this theorem is implying that change looks the same from all directions. But even if this is true, I would like to know how this can be visualised.
Or tell me if this is really not important and I should just trust what the theorem says?
Thanks
Roughly speaking, derivatives are "very small increments" of functions: $$\frac{\partial f}{\partial x}\approx \frac{f(x+\epsilon, y)-f(x, y)}{\epsilon}=\frac{\Delta_x f}{\epsilon},$$ and $$\frac{\partial f}{\partial y} \approx \frac{f(x, y+\epsilon)-f(x, y)}{\epsilon}=\frac{\Delta_y f}{\epsilon}.$$ Now clearly $\Delta_x\Delta_y f=\Delta_y\Delta_x f$. Therefore, from the relation $$ \frac{\partial^2 f}{\partial x\partial y}\approx\frac{\Delta_x\Delta_y f}{\epsilon^2}=\frac{\Delta_y \Delta_x f}{\epsilon^2}\approx\frac{\partial^2 f}{\partial y\partial x}, $$ one expects that partial derivatives commute as well. That this is not always the case is due to the fact that some continuity is needed in order to pass to the limit and convert the $\approx$ relations into precise identities.