Boosting Vector Calculus with the Graphical Notation

TL;DR

This paper introduces a graphical notation for three-dimensional Euclidean vector calculus to improve teaching and learning, making the subject more intuitive and reducing reliance on bulky index notation.

Contribution

It pioneers the application of graphical tensor notation to vector calculus, providing a new pedagogical tool for physics education.

Findings

Graphical notation simplifies vector calculus learning.

Students show increased engagement with the graphical approach.

The method enhances understanding of tensor language in physics.

Abstract

Learning vector calculus techniques is one of the major missions to be accomplished by physics undergraduates. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Meanwhile, there have been graphical notations for tensor algebra that are intuitive and effective in calculations and can serve as a quick mnemonic for algebraic identities. Although they have been introduced and applied in vector algebra in the educational context, to the best of our knowledge, there have been no publications that employ the graphical notation to three-dimensional Euclidean vector calculus, involving differentiation and integration of vector fields. Aiming for physics students and educators, we introduce such "graphical vector calculus," demonstrate its pedagogical advantages, and provide enough exercises containing both purely mathematical identities and practical calculations in physics. The graphical notation can readily be utilized in the educational environment to not only lower the barriers in learning and practicing vector calculus but also make students interested and self-motivated to manipulate the vector calculus syntax and heuristically comprehend the language of tensors by themselves.