Longitudinal and transverse components of a vector field

TL;DR

This paper provides a pedagogical overview of the longitudinal and transverse components of vector fields, clarifying their relation to the Helmholtz theorem and applicability to both time-dependent and independent fields.

Contribution

It offers a unified, pedagogical account of Belinfante's projective delta functions and their connection to the Helmholtz decomposition for vector fields.

Findings

Clarifies the relation between Belinfante's delta functions and Helmholtz theorem

Demonstrates applicability to both time-dependent and time-independent fields

Provides a unified pedagogical perspective

Abstract

A unified account, from a pedagogical perspective, is given of the longitudinal and transverse projective delta functions proposed by Belinfante and of their relation to the Helmholtz theorem for the decomposition of a three-vector field into its longitudinal and transverse components. It is argued that the results are applicable to fields that are time-dependent as well as fields that are time-independent.