Proofs of Vector Identities Using Tensors

TL;DR

This paper presents proofs of various vector identities using tensor calculus, including some identities proved for the first time, aiding students in understanding vector algebra in physics.

Contribution

It introduces new proofs of vector identities using Levi-Civita symbols and Kronecker delta tensors, some of which are novel contributions.

Findings

Several vector identities proved using tensor methods

Some identities are proved for the first time

The derivations are suitable for students at different levels

Abstract

The vector algebra and calculus are frequently used in many branches of Physics, for example, classical mechanics, electromagnetic theory, Astrophysics, Spectroscopy, etc. Important vector identities with the help of Levi-Civita symbols and Kronecker delta tensor are proved and presented in this paper. Some of the identities have been proved using Levi-Civita Symbols by other mathematicians and Physicists. The rests are presented for the first time. The derivations are of interest for both graduate and undergraduate students.