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Clairaut's theorem

In mathematical analysis, Clairaut's theorem states that if

f \colon \mathbb{R}^n \to \mathbb{R}

has continuous second partial derivatives at (a_1, \dots, a_n), then for 1 \leq i,j \leq n,

\frac{\partial^2 f}{\partial x_i\, \partial x_j}(a_1, \dots, a_n) = \frac{\partial^2 f}{\partial x_j\, \partial x_i}(a_1, \dots, a_n).

In words, the partial derivatives of this function commute. This theorem is named for the French mathematician Alexis Clairaut.

See also

Last updated: 05-27-2005 03:42:46
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