## You are here

HomeClairaut's theorem

## Primary tabs

# Clairautβs theorem

###### Clairautβs 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier