Recovering seldom-used theorems of vector calculus and their application to problems of electromagnetism

TL;DR

This paper employs differential forms to prove rarely documented theorems in vector calculus, introduces new theorems, and demonstrates their applications in electromagnetism, providing novel insights and methods for physics problems.

Contribution

It recovers seldom-used theorems of vector calculus using differential forms, including two previously unpublished theorems, and applies them to electromagnetism problems.

Findings

New proofs of integral vector calculus theorems

Introduction of two previously unpublished theorems

Enhanced methods for computing forces and torques in electromagnetism

Abstract

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to problems in physics are shared including a general approach for computing net forces and torques on current-carrying loops that yields insights that are not evident from the standard approach.