# Clairaut’s theorem

###### Clairaut’s Theorem.

If $\mathrm{f}\mathrm{:}{\mathrm{R}}^{n}\mathrm{\to}{\mathrm{R}}^{m}$ is a function whose second partial derivatives^{} exist and are continuous^{} on a set $S\mathrm{\subseteq}{\mathrm{R}}^{n}$, then

$$\frac{{\partial}^{2}f}{\partial {x}_{i}\partial {x}_{j}}=\frac{{\partial}^{2}f}{\partial {x}_{j}\partial {x}_{i}}$$ |

on $S$, where $\mathrm{1}\mathrm{\le}i\mathrm{,}j\mathrm{\le}n$.

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 $\mathbf{F}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ is a function satisfying the hypothesis^{}, then $\nabla \cdot (\nabla \times \mathbf{F})=0$.
Or, if $f:{\mathbb{R}}^{3}\to \mathbb{R}$ is a function satisfying the hypothesis, then $\nabla \times \nabla f=\mathrm{\U0001d7ce}$.

Title | Clairaut’s theorem |
---|---|

Canonical name | ClairautsTheorem |

Date of creation | 2013-03-22 13:53:44 |

Last modified on | 2013-03-22 13:53:44 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 18 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 26B12 |

Synonym | equality of mixed partials |