For starters, such a thing as the divergence of the normal vector is already "counter intuitive" to me because of the definition of divergence I like using (local flux density, involving an infinitesimal volume in $\mathbb{R}^3$, whereas the normal vector is only defined on a surface)
I've been introduced to the title's equality in the context of Laplace's law for pressure difference across an interface. I think I would need first an interpretation of "divergence of normal vector".
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