Clairaut’s theorem
Clairaut’s Theorem.
If is a function whose second partial derivatives exist and are continuous
on a set , then
on , where .
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 is a function satisfying the hypothesis, then .
Or, if is a function satisfying the hypothesis, then .
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 |