A Stokes theorem for everyone

TL;DR

This paper presents a simplified, general version of Stokes' theorem applicable to continuous differential forms on compact sets, making the theorem more accessible without sacrificing mathematical rigor.

Contribution

It introduces a straightforward formulation of Stokes' theorem that requires less advanced machinery, broadening its usability in practical contexts.

Findings

The theorem applies to continuous forms on compact sets built of 'bricks'.

It does not require orientability for the application of the theorem.

The proof uses integration by parts and inner approximation techniques.

Abstract

Many versions of the Stokes theorem are known. More advanced of them require complicated mathematical machinery to be formulated which discourages the users. Our theorem is sufficiently simple to suit the handbooks and yet it is pretty general, as we assume the differential form to be continuous on a compact set F(A) and C1 "inside" while F(A) is built of "bricks" and its inner part is a C1 manifold. There is no problem of orientability and the integrals under consideration are convergent. The proof is based on integration by parts and inner approximation.