4.10 More on partial derivatives  (Page 2/3)

Suppose $S$ is a subset of $R^2,$ and that $f$ is a continuous real-valued function on $S.$ If both partial derivatives of $f$ exist at each point of the interior $S^0$ of $S,$ and both are continuous on $S^0,$ then $f$ is said to belong to $C^1\left(S\right).$ If all $k$ th order mixed partial derivatives exist at each point of $S^0,$ and all of them are continuous on $S^0,$ then $f$ is said to belong to $C^k\left(S\right).$ Finally, if all mixed partial derivatives, of arbitrary orders, exist and are continuous on $S^0,$ then $f$ is said to belong to $C^{\infty }\left(S\right).$

Perhaps the most interesting theorem about partial derivatives is the “mixed partials are equal” theorem. That is, $f_{xy}=f_{yx}.$ The point is that this is not always the case. An extra hypothesis is necessary.

Theorem on mixed partials

Let $f:S\to R$ be such that both second order partials derivatives $f_{xy}$ and $f_{yx}$ exist at a point $(a,b)$ of the interior of $S,$ and assume in addition that one of these second order partials exists at every point in a disk $B_r(a,b)$ around $(a,b)$ and that it is continuous at the point $(a,b).$ Then $f_{xy}(a,b)=f_{yx}(a,b).$

Suppose that it is $f_{yx}$ that is continuous at $(a,b).$ Let $ϵ>0$ be given, and let ${\delta }_1>0$ be such that if $|(c,d)-(a,b)|<{\delta }_1$ then $|f_{yx}(c,d)-f_{yx}(a,b)|<ϵ.$ Next, choose a ${\delta }_2$ such that if $0<\left|k\right|<{\delta }_2,$ then

and fix such a $k.$ We may also assume that $\left|k\right|<{\delta }_1/2.$ Finally, choose a ${\delta }_3>0$ such that if $0<\left|h\right|<{\delta }_3,$ then

and

and fix such an $h.$ Again, we may also assume that $\left|h\right|<{\delta }_1/2.$

In the following calculation we will use the Mean Value Theorem twice.

because $b^{\text{'}}$ is between $b$ and $b+k,$ and $a^{\text{'}}$ is between $a$ and $a+h,$ so that $|(a^{\text{'}},b^{\text{'}})-(a,b)|<{\delta }_1/\sqrt{2}<{\delta }_1.$ Hence, $|f_{xy}(a,b)-f_{yx}(a,b)<4ϵ,$ for an arbitrary $ϵ,$ and so the theorem is proved.