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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

$$\frac{\partial ^2 f}{\partial x \, \partial y} = \frac{\partial^2 f}{\partial y \, \partial x}$$

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    [A (perversely) related question.](http://math.stackexchange.com/questions/104735)2012-07-14
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    [This](http://www.math.ubc.ca/~feldman/m263/mixed.pdf) link has a nice proof, and probably is the "consider this function" proof you have in your book, but the remark does a good job of saying why we consider such a function, and brings out the heart of the problem. Usually the intuitive part is that they should be equal, and the bit that denies us is that they can be unequal. Equal of mixed partials comes down to the ability to interchange a pair of limits and when you see this problem in this light it is easier to construct a counterexample.2012-07-14

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