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2.14 Equality of mixed partials

Note that $f_{xy}=f_{yx}$ in the example from the previous slide. More generally:

Theorem.

If $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ has the property that the second-order mixed partial derivatives $f_{xy}$ and $f_{yx}$ exist and are continuous, then $f_{xy}=f_{yx}$ .

This result is a great time-saver. Instead of working out mixed partial derivatives in all possible orders, it is enough to differentiate in the most convenient order.