There is a theorem stating the following:(adsbygoogle = window.adsbygoogle || []).push({});

Let f be defined in a neighborhood of (x0, y0) in R^2. Suppose f has partial derivatives f_1, f_2, f_12, and f_21 in this neighborhood and that the cross partials f_12 and f_21 are continuous at (x0, y0). Then the cross partials f_12 and f_21 are equal at (x0, y0).

This is a theorem proven in TBB's Elementary Real Analysis. It is stated that there are other conditions that could be met instead. I will reqrite the theorem with this other condition:

Let f be defined in a neighborhood of (x0, y0) in R^2. Suppose f has partial derivatives f_1, f_2, f_12, and f_21 in this neighborhood and thatthe partials f_1 and f_2 are differentiable at (x0, y0). Then the cross partials f_12 and f_21 are equal at (x0, y0).

How would you prove this or start a proof of this? No, this is not a homework problem, this is a theorem I'm curious about myself for my own satisfaction. Also, if my notation is not clear, f_1 is a partial derivative with respect to x, f_2 is the partial derivative with respect to y.

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# Proving that cross partials in R^2 are equal if

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