Let Ω be the region in $ℝ^3$ defined by $$ Ω={(x_1,x_2,x_3):max(∣∣x_1∣∣,∣∣x_2∣∣,∣∣x_3∣∣)≤1}$$ Let ∂Ω denote the boundary of Ω.
Calculate $$∫_{∂Ω}ϕF⋅ndσ$$
where n is the unit normal vector, dσ denotes integration over ∂Ω,
$F_i=\large \frac{x_i}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}}^=\frac{x_i}{r^3}$
and $ϕ(y_1,y_2,y_3)$ is a continuously differentiable function of $\large y_i=\frac{x_i}{r}$. Assume that ϕ has unit mean over the unit sphere.
I just started on this problem so I don't want solutions.
My question is: how should I interpret $\phi F$? Is $\phi$ another vector field and that I should take the inner product of $\phi$ with $F$?
This wouldn't make much sense since; I'd end up with a scalar, and then scalar.$\vec n$ wouldn't really make sense either.
Any hints or suggestions are welcome.
Thanks,
EDIT: I'd welcome solutions at this point. I am getting weird computations -- such as an integral that is equal to zero. I tried using the "product rule" that I found on Wolfram Alpha to compute the divergence of $\phi F$. I notice first that divF=0, so F alone is divergence-free. But I honestly do not know whether I have the correct vector field after multiplication with $\phi$. So, when computing the divergence of $\phi F$, I might be using an incorrect vector field.