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I have an intuitive understanding of the gradient, divergence, curl, and Laplacian operators in multivariable calculus, but not of the vector Laplacian. Is there a visualizable intuition for the meaning of the Laplacian of a vector field on a Riemannian manifold?

And a closely related question: is there an intuitive explanation for what's "special" about solutions to the vector Laplace equation?

tparker
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    Without a relation between the components, it's just the Laplacian of a few unrelated scalar fields. You need an additional relationship between the components for it to have some deeper significance. – Ian Dec 06 '20 at 17:40
  • @Ian I see what you're saying but I disagree. The components of a vector field aren't scalars, nor are they unrelated; they intermix in well-defined ways upon coordinate rotations (which is exactly what makes them geometric quantities). Only in Cartesian coordinates does the Laplacian decouple - and that only works on Euclidean space, while the vector Laplacian is defined on an arbitrary pseud-Riemannian manifold. – tparker Dec 06 '20 at 18:52
  • Just saying that the vector field is transformed in the same way as the ambient space is already imposing more geometry than you set up in the original question. Not trying to be rude, I am just saying that the insight you're looking for probably needs more structure than you have incorporated into the question. – Ian Dec 06 '20 at 19:12
  • @Ian Isn't that fact built into the definition of a tensor field, of which a vector field is a special case? I'm not trying to be rude either, but I don't think you quite understand what a vector field is if you think that it just assigns an ordered tuple to each point in space. – tparker Dec 06 '20 at 19:46
  • In case it helps, I've edited the question to specify that I'm talking about a vector field on a Riemannian manifold. – tparker Dec 06 '20 at 19:48

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I found an answer. The property discussed at https://math.stackexchange.com/a/50285/268333 generalizes easily to the vector case for vector fields on $\mathbb{R}^d$:

$$ \nabla^2\, {\bf F}({\bf x}_0) = 2d \lim_{r \to 0^+} \frac{\langle {\bf F} \rangle_r - {\bf F}({\bf x}_0)}{r^2} , $$

where $\langle {\bf F} \rangle_r$ denotes the average value of the vector field over the small sphere of radius $r$ centered at ${\bf x}_0$. This means that the vector Laplacian of a vector field at a point is proportional to the leading-order amount by which the average value of the vector field near the point differs from its value at the point.

tparker
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