PlanetMath: Clairaut's theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Clairaut's theorem

(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

$\displaystyle \frac{\partial^2 f}{\partial x_i \partial x_j} =\frac{\partial^2 f}{\partial x_j \partial x_i}$

on , 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 $\nabla \times \nabla f= \mathbf{0}$ .

"Clairaut's theorem" is owned by Mathprof. [ full author list (3) | owner history (2) ]

(view preamble)

Other names:

equality of mixed partials

Log in to rate this entry.
(view current ratings)

Cross-references: hypothesis, curl, divergence, gradient, properties, Calculus, vector, continuous, partial derivatives, function
There are 2 references to this entry.

This is version 15 of Clairaut's theorem, born on 2003-08-22, modified 2006-11-13.
Object id is 4642, canonical name is ClairautsTheorem.
Accessed 6564 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact