Vector calculus in two-dimensional space

TL;DR

This paper provides a detailed pedagogical account of two-dimensional vector calculus using differential forms, filling a gap in educational resources for this less common but important mathematical area.

Contribution

It introduces a comprehensive, accessible presentation of 2D vector calculus based on differential forms, bridging a gap in existing educational materials.

Findings

Clarifies the theory of 2D vector calculus in standard notation

Connects 2D calculus with differential forms for better understanding

Provides detailed explanations suitable for learners

Abstract

Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is based on the logic of theory of differential forms. For readers not familiar with the latter, the results are presented in detail in the standard language and notations.