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Given a smooth parametric surface $S(u,v)$ and a tangential vector field $F$ on it, e.g. the surface gradient of a function $f$ on the surface as I visualised here (drag the middle mouse button to rotate the surface, press R to generate a new random function and press E or V to toggle displaying edges or vertices), I'm looking for an intuitive view on the divergence of such a vector field (also known as the surface- or tangential divergence).

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For a point $P$ on the surface, I suppose the first step is to consider its tangent plane as well as some (orthonormal) coordinate system $(r,s)$ on it. But what's next? Our vector field doesn't live on this tangent plane, but on the surface — we cannot simply compute, say, $\frac{\partial}{\partial r} F_r + \frac{\partial}{\partial s} F_s$. Although, for a surface that's isometric to the plane, I suppose we could actually do that (i.e. gently unfold $S$ onto the plane without stretching/compressing it and compute the divergence of the resulting 2D vector field), but that's hardly the general case.

Then, what about using one of the alternative definitions of divergence? Perhaps a curve- or surface integral limit on the surface around $P$... But that doesn't sound very pragmatic — I'd like to be able to actually compute values. Other definitions of divergence use e.g. differential forms, though I don't really understand these in the context of a surface (in $\mathbb{R}^3$ the divergence corresponds to a 3-form, but on a surface we only have two linearly independent tangent vectors to play with). I'm currently reading up on this topic, so I might get back to this point. Then there's the notion of the covariant derivative, which I understand in theory (i.e. using parallel transport to move a tangent vector to a different tangent plane) but I'm not sure how to apply it (i.e. implement it) in practise.

Perhaps there is some value in the idea of unfolding a surface onto the plane. Now going with the general case, this involves locally stretching/compressing the surface, which is where the first fundamental form (i.e. the metric) comes in. Another way to look at this would be to 'pull back' the tangent vectors in some neighbourhood around $P$ to the $(u,v)$ domain (essentially using the inverse of Jacobian, which maps vectors in the $(u,v)$ domain to tangent vectors on $S$), compute the divergence of that 2D vector field, and finally somehow correct the obtained value. I'm not quite sure how that would work though, and I'm hoping someone here has some good insight.

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For two-dimensional surfaces the most intuitive approach I can think for computing divergence is to adopt an isothermal system of local coordinates on the surface (These are two functions $(u,v)$ whose gradient vectors on the surface are perpendicular and have equal length $\lambda$ at any given point. ) Then the metric is especially simple: $ds^2= \lambda^2 (du^2+ dv^2)$. This is a special case of an orthogonal system of coordinates.

The divergence of a vector field $\vec F$ is simply the flux of the vector field across the edges of a small cell, divided by the area of that cell. Take the cell to be bounded by contours of $u$ and $v$. Since this is an orthogonal system of coordinates, you can just look up how divergence is computed in such coordinate systems.

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  • $\begingroup$ Although this sounds potentially useful (reminds me a bit of the principal directions on a surface), I don't immediately see the advantage it would bring to computing the surface divergence. Do you have any references and/or examples of using such an isothermal system in this context? $\endgroup$
    – Ailurus
    Commented Feb 9, 2023 at 11:05

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