Mean of vector field using info at surface

[madypa]

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!

 

[jvicens]

Thank you for your question. It seems that you are on the right track with your approach. However, there are a few things to consider in order to calculate the mean of U in this closed surface.

Firstly, it is important to note that the mean of U cannot be calculated solely based on the information along the surface. It also requires knowledge of the vector field U within the volume enclosed by the surface. This is because the mean of U is a volume average, not just a surface average.

Secondly, as you mentioned, the divergence of U is zero. This means that the volume integral of the divergence of U over the enclosed volume is also zero. This can be seen from the divergence theorem, which relates the volume integral of the divergence of a vector field to the surface integral of the dot product of the vector field and the surface normal.

Therefore, using the divergence theorem, we can rewrite the equation as:

∇. mean(U) = 1/V ∫_A U . n dA = 0

This means that the mean of U is a constant value, and it is equal to zero. This may seem counterintuitive, but it is a consequence of the given information (i.e. the divergence of U is zero).

In conclusion, the mean of U in this closed surface is zero. I hope this helps to clarify your calculation. Let me know if you have any further questions or concerns.