Clairaut's Theorem

f \colon \mathbb{R}^n 	o \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,

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

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\{AP
ight\}=\cos\!\left\{Lat
ight\}\sin\!\left\{Az
ight\}.\,\!

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Information About

Clairaut's Theorem