Geometric definition of divergence using curvature mentioned in Tristan Needham

Question

In page-479 of Visual Complex Analysis, Tristan Needham derives the flux of a vector field in Geometric form:

$$ \nabla \cdot X = \partial_s |X| + \kappa_p |X|$$

The $\partial_s$ is a derivative along streamlines of the vector field $X$ an $\kappa_p$ is the curvature of the orthogonal streamline at the point we are taking divergence at.

Where did this formula originate from, and, what other sources discuss it? There is nothing of it on wikipedia. The book itself claims it couldn't find any reference to the formula in Modern literature, but is it still true in 2022 ?

Answer 1

Given any hypersurface of a Riemannian manifold with a unit normal vector field $\nu$, extend $\nu$ to be unit length. Then $$\operatorname{div}(u\nu)=u\operatorname{div}\nu+\text{d}u(\nu).$$ This proves that if $X$ is any nonvanishing vector field on a neighborhood of the hypersurface, which is orthogonal to the hypersurface, then $$\operatorname{div}X=|X|H+\text{d}|X|(\nu)$$ where $H$ is the mean curvature. And so, when only given a Riemannian manifold, this holds also when $X$ is a hypersurface-orthogonal vector field (also called "complex lamellar vector field"); $H$ is the mean curvature of the orthogonal hypersurface and $\nu$ is $\frac{X}{|X|}$. Every vector field on a two-dimensional space is hypersurface-orthogonal, so this fully contains Needham's formula as a special case.