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# 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 $\nabla\times\nabla f=\mathbf{0}$.

Synonym:

equality of mixed partials

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

26B12*no label found*

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