What was given in my calc book is a "consider the function" proof. That is, the author gives a function out of the blue and would deduce all the nice properties from it. I'd prefer a proof which is motivated (perhaps, intuitive) - you see how the proof is crafted in the mind of the person. So my question is a geometric or intuitive proof of
$\frac{\partial ^2 f}{\partial x \, \partial y} = \frac{\partial^2 f}{\partial y \, \partial x}$