How to explain Clairaut-Schwartz's Theorem, $f_{xy}=f_{yx}$? I am looking for a non-technical explanation of Clairaut's theorem which states that the mixed derivative of smooth functions are equal.
A geometrical, graphical, or demo that explains the theorem and its implications will be helpful. I am not looking for a proof!

I don't think that there is a simple geometric or physical argument that makes this theorem intuitively obvious. In the following I try to explain what kind of information about $f$ the mixed partials do encode. That they are equal then follows from symmetry.
Consider the small square $Q_h:=[-h,h]^2$ and add up the values of $f$ at the vertices of $Q_h$, albeit with alternating signs:
$$\Phi(h):=f(h,h)-f(-h,h)+f(-h,-h)-f(h,-h)\ .$$
Assume that the mixed partial derivatives of $f$ are defined and continuous in a neighborhood of  $(0,0)$. Then we can write
$$\eqalign{\Phi(h)&=\int_{-h}^h f_x(t,h)\>dt-\int_{-h}^h f_x(t,-h)\>dt=\int_{-h}^h\int_{-h}^h f_{xy}(t,s)\>ds\>dt\cr &= f_{xy}(\tau,\sigma)|Q_h|\cr}$$
for some point $(\tau,\sigma)\in Q_h$. Doing the same calculation the other way around we obtain
$$\Phi(h)= f_{yx}(\tau',\sigma')|Q_h|$$
for some point $(\tau',\sigma')\in Q_h$. Letting $h\to0+$ the claim follows.
