Clairaut's theorem

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In mathematical analysis, Clairaut's theorem states that if

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

has continuous second partial derivatives at any given point in $\mathbb{R}^n$ , say, $(a_1, \dots, a_n),$ 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 at that point. This theorem is named after the French mathematician Alexis Clairaut.

[edit] Clairaut's constant

A byproduct of this theorem is Clairaut's constant (alternatively known as "Clairaut's formula" and "Clairaut's parameter"), which relates the latitude, $\phi\,\!$ , and (here, spherical) azimuth, $\widehat{\alpha}\,\!$ , of points on a sphere's great circle. The identification of a particular great circle equals its azimuth at the equator, or arc path, $\widehat{\Alpha}\,\!$ :