Clairaut’s theorem

Clairaut’s Theorem.

If f:RnRm is a function whose second partial derivativesMathworldPlanetmath exist and are continuousMathworldPlanetmathPlanetmath on a set SRn, then


on S, where 1i,jn.

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 𝐅:33 is a function satisfying the hypothesisMathworldPlanetmath, then (×𝐅)=0. Or, if f:3 is a function satisfying the hypothesis, then ×f=𝟎.

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