On the Integrability of Pfaffian Forms on ${\mathbb R}^{n}$

Abstract

This paper details the lesser known conditions on ${\mathbb {R}}^{n}$ for the integrability of pfaffian forms, or 1-forms. Emphasis is given to locality of these conditions, and proofs in some additional detail are provided for theorems due to Clairaut and Carath\'eodory. Considering the importance of the integrability of pfaffian forms, in particular in mathematical-physics, this paper shows that: there is a hidden content in Carath\'eodory's theorem in the direction of a global integrability.