Clairaut's theorem

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In mathematical analysis, Clairaut's theorem states that if

[f \colon \mathbb^n \to \mathbb]

has continuous second partial derivatives at any given point in [ \mathbb^n ], say, [ (a_1, \dots, a_n),] then for [1 \leq i,j \leq n,]

[\frac(a_1, \dots, a_n) = \frac(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.

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 (Lat) and azimuth (Az) of points on a sphere's great circle. The identification of a particular great circle equals its azimuth at the equator, or arc path (AP):

[\sin\!\left\=\cos\!\left\\sin\!\left\.\,\!]

See also

• Symmetry of second derivatives
• Fubini's theorem

External link

• [Clairaut's formula]

 

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