# Integration of a solenoidal vector field over a volume

#### Phillips101

1. The problem statement, all variables and given/known data

div(J)=0 in volume V, and J.n=0 on surface S enclosing V, where n is the normal vector to the surface.

Show that the integral over V of J dV is zero.

2. Relevant equations

3. The attempt at a solution

I can't get anywhere with it! The divergence theorem doesn't seem to help, as I just go round in circles. Any help at all will really be useful.

Thanks

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#### gabbagabbahey

Homework Helper
Gold Member
Try applying the divergence theorem to the vector field $$\lambda\vec{J}$$ What happens if you choose $\lambda$ such that $$\vec{\nabla}\lambda$$ is a constant vector?

#### Phillips101

Thanks for that - everything now works.

"Integration of a solenoidal vector field over a volume"

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