Editing Symmetry of second derivatives (section)

== Application to differential forms ==
The Clairaut-Schwarz theorem is the key fact needed to prove that for every <math>C^\infty</math> (or at least twice differentiable) [[differential form]] <math>\omega\in\Omega^k(M)</math>, the second exterior derivative vanishes: <math>d^2\omega := d(d\omega) = 0</math>.  This implies that every differentiable [[exact differential form|exact]] form (i.e., a form <math>\alpha</math> such that <math>\alpha = d\omega</math> for some form <math>\omega</math>) is [[Closed differential form|closed]] (i.e., <math>d\alpha = 0</math>), since <math>d\alpha = d(d\omega) = 0</math>.{{sfn|Tu|2010}}

In the middle of the 18th century, the theory of differential forms was first studied in the simplest case of 1-forms in the plane, i.e. <math>A\,dx + B\,dy</math>, where <math>A</math> and <math>B</math> are functions in the plane. The study of 1-forms and the differentials of functions began with Clairaut's papers in 1739 and 1740. At that stage his investigations were interpreted as ways of solving [[ordinary differential equation]]s. Formally Clairaut showed that a 1-form <math>\omega = A \, dx + B \, dy</math> on an open rectangle is closed, i.e. <math>d\omega=0</math>, if and only <math>\omega</math> has the form <math>df</math> for some function <math>f</math> in the disk. The solution for <math>f</math> can be written by Cauchy's integral formula

:<math>f(x,y)=\int_{x_0}^x A(x,y)\, dx + \int_{y_0} ^y B(x,y)\, dy;</math>

while if <math> \omega= df</math>, the closed property <math> d\omega=0</math> is the identity <math>\partial_x\partial_y f = \partial_y\partial_x f</math>. (In modern language this is one version of the [[Poincaré lemma]].){{sfn|Katz|1981}}