The value of curl(curl A) - grad(div A) + div(grad A) for an absolute vector A

TL;DR

This paper clarifies the mathematical identity involving curl, div, and grad operators for an absolute vector in both Riemannian and Euclidean spaces, emphasizing the roles of Riemann and Ricci tensors.

Contribution

It provides a rigorous, coordinate-independent analysis of the identity, resolving common misconceptions in literature and textbooks.

Findings

Clarifies the identity in Riemannian and Euclidean contexts

Highlights the intrinsic role of Riemann and Ricci tensors

Resolves confusion in existing mathematical literature

Abstract

The well-known identity involving the expression presented in the above title is considered in Riemannian and in Euclidean space without restriction on the coordinate system adopted therein. The Riemann and Ricci tensors intrinsically assume a defining role in the analysis. The analysis is designed to put an end to the myriad of confusing and mostly incorrect statements about the identity, which are found in textbooks and in the literature.