Clairaut's theorem
ClairautsTheorem
2003-08-22 03:21:04
2006-11-13 15:21:39
Theorem
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\newtheorem*{cthm*}{Clairaut's Theorem}
\begin{cthm*}
If $\mathbf{f}\colon\mathbb{R}^n \to \mathbb{R}^m$ is a function whose second partial derivatives exist and are continuous on a set $S \subseteq \mathbb{R}^n$, then
\[
\frac{\partial^2 f}{\partial x_i \partial x_j}
=\frac{\partial^2 f}{\partial x_j \partial x_i}
\]
on $S$, where $1 \leq i,j \leq n$.
\end{cthm*}
This theorem is commonly referred to as \emph{the equality of mixed partials}.
It is usually first presented in a vector calculus course,
and is useful in this context for proving basic properties of the interrelations of gradient, divergence, and curl.
For example, if $\mathbf{F}\colon \mathbb{R}^3 \to \mathbb{R}^3$ is a function satisfying the hypothesis, then $\nabla \cdot (\nabla \times \mathbf{F}) =0$.
Or, if $f\colon\mathbb{R}^3 \to \mathbb{R}$ is a function satisfying the hypothesis, then $\nabla \times \nabla f= \mathbf{0}$.