Introducing the notion of tensors through a variation of a Feynman didactic approach

TL;DR

This paper offers a simplified, visual, and accessible approach to teaching tensors, inspired by Feynman's didactic method, using atoms and molecules instead of crystals and reducing mathematical prerequisites.

Contribution

It introduces a new didactic framework for tensors based on atomic and molecular models, with visual tools, making tensor concepts more accessible to beginners.

Findings

Reduces mathematical prerequisites for tensor understanding.

Provides visual representations based on stress ellipsoids.

Applicable as a teaching resource for early undergraduate students.

Abstract

In one of his books [$\textit{The Feynmann Lectures on Physics}$, vol. 2], Feynman presents a didactic approach to introduce basic ideas about tensors, using, as a first example, the dependence of the induced polarization of a crystal on the direction of the applied electric field, and also presenting the energy ellipsoid as a way of visualizing the polarization tensor. In the present paper, we propose some variations on Feynman's didactic approach, considering as our basic models a single ground-state atom and a carbon dioxide ($\text{CO}_{2}$) molecule, instead of crystals, and introducing a visual representation of tensors based on the ideas of the Lam\'e stress ellipsoid, instead of the energy ellipsoid. With these changes, the resulting didactic proposal presents a reduction in the prerequisites of physical and mathematical concepts if compared to Feynman's original approach, requiring, for example, no differential calculus and only introductory vector algebra. The text is written so that it can be used directly as a learning tool for students (even those in the beginning of the undergraduate course), as well as for teachers interested in preparing their own materials.