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

TL;DR

This paper explores the conditions under which Pfaffian forms on Euclidean space are integrable, emphasizing local criteria and revealing a hidden aspect of Carathéodory's theorem related to global integrability, with implications for mathematical physics.

Contribution

It provides detailed proofs and insights into the local and global integrability conditions of Pfaffian forms, highlighting a new perspective on Carathéodory's theorem.

Findings

Local integrability conditions are clarified.

A hidden aspect of Carathéodory's theorem is identified.

Implications for mathematical physics are discussed.

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.