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

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.