Clairaut's theorem
From Wikipedia, the free encyclopedia
In mathematical analysis, Clairaut's theorem or Schwarz's theorem, named after Alexis Clairault and Hermann Schwarz, states that if
has continuous second partial derivatives at any given point in 
, say, 
then for 
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, 
, and (here, spherical) azimuth, 
, of points on a sphere's great circle. The identification of a particular great circle equals its azimuth at the equator, or arc path, 
:
