The 3 stooges of vector calculus and their impersonators: A viewer's guide to the classic episodes

TL;DR

This paper offers a combinatorial perspective on vector calculus theorems, replacing traditional operators with substitutes, simplifying proofs, and introducing cohomology theory for a deeper understanding.

Contribution

It introduces combinatorial substitutes for Grad, Div, and Curl, providing simpler proofs of key theorems and a brief introduction to cohomology theory.

Findings

Simplified proofs of Green's theorem

Equivalence of integral and derivative definitions of curl

Introduction to cohomology theory

Abstract

The basic theorems of vector calculus are illuminated when we replace the original 3 stooges of vector calculus: Grad, Div, and Curl, with combinatorial substitutes. In addition to providing simple proofs of Green's theorem and the equivalence of the integral and derivative definitions of curl, we also provide a brief introduction to cohomology theory.