Tensor-based derivation of standard vector identities

TL;DR

This paper introduces a tensor-based method to derive standard vector identities symbolically, avoiding component expansions and providing a new teaching approach for physics and engineering students.

Contribution

It presents a novel tensor-based derivation technique for vector identities, enhancing conceptual understanding without relying on components or geometrical arguments.

Findings

Derivations performed using skew-symmetric tensors and dyadic products

Simplifies teaching of vector identities in physics

Provides examples of physical skew-symmetric tensors

Abstract

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are usually proved using Cartesian components or geometrical arguments, accordingly. Instead, this work presents a new teaching strategy in order to derive symbolically vector identities without analytical expansions in components, either explicitly or using indicial notation. This strategy is mainly based on the correspondence between three-dimensional vectors and skew-symmetric second-rank tensors. Hence, the derivations are performed from skew tensors and dyadic products, rather than cross products. Some examples of skew-symmetric tensors in Physics are illustrated.