Let $V = \left( (x,y,z) \in R^3 : 1/4 \le x^2 + y^2 + z^2 \le 1\right)$
and $\vec{F} = \dfrac{x\hat{i} + y\hat{j} + z\hat{k}}{ (x^2 + y^2 +z^2)^2}$ for $(x,y,z)\in V$ Let $\hat{n}$ denote the outward unit normal vector to the boundary of V and $S$ denote the part $\left(x^2+ y^2 + z^2 = 1/4\right)$ of the boundary of $V$
Then find $\displaystyle \int \int F.\hat{n} dS$
I want to evaluate this surface integral with help of Gauss Divergence theorem
However ,my main issue is which surface should I use ?From what I have learnt the body used for Gauss Divergence theorem must be the outer boundary for surface.
So, in this case If I take the outer boundary itself to be $x^2 +y^2 + z^2 =1/4$ then ,
$\vec{F} = \dfrac{\vec{r}}{r^4}$ So $\nabla .F = \dfrac{-1}{r^4}$
Using this the integral is given by : (After simplifying in terms of Spherical Coordinates)
$\displaystyle \int_{0}^{2\pi} \int_{0}^{\pi}\int_{0}^{1/2} 1/{{\rho}^2} sin \phi d \rho d \phi d \theta$ .
However this last integral is undefined .
Can anyone tell me what is the correct way to solve this question
Thank You.