# Mean of vector field using info at surface

1. May 5, 2012

Hello,

I have a 3d closed surface. This closed surface lies in a 3d vector field. I know the value of the vector at discrete points along the surface and the surface normal at these points. Essentially, say vector U and vector n at these points. This is the only information that I have. I need the *volume* mean of U in this closed surface. I also know that the divergence of U = 0. Seems trivial, but I can't seem to get through the calculation. Here is what I did:

mean(U) = 1/V ∫ U dV

where V is the volume enclosed by the closed surface.

Lets take the divergence on both sides, assuming mean U and U are continuously differentiable and the divergence and integration operators can be interchanged.

∇. mean(U) = 1/V ∫ ∇. U dV
= 1/V ∫_A U . n dA

So I can connect the divergence of mean of U with the information along the surface. However, I need the mean(U).

Any thoughts?

Thanks!