# Clairaut’s theorem

###### Clairaut’s Theorem.

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 $1\leq i,j\leq n$.

This theorem is commonly referred to as 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 $\nabla\times\nabla f=\mathbf{0}$.

Title Clairaut’s theorem ClairautsTheorem 2013-03-22 13:53:44 2013-03-22 13:53:44 Mathprof (13753) Mathprof (13753) 18 Mathprof (13753) Theorem msc 26B12 equality of mixed partials