Surface integral over a discontinuous flux

Relevant wiki: surface integral

First we need to evaluate the divergent of $F(x,y,z)$ , making:

$\nabla \cdot F(x,y,z)=\frac{xy(4-\sqrt{x^{2}+y^{2}})}{\sqrt{((4-\sqrt{x^{2}+y^{2}})^2+z^2)}^{3}\sqrt{x^{2}+y^{2}}} -\frac{yx(4-\sqrt{x^{2}+y^{2}})}{\sqrt{((4-\sqrt{x^{2}+y^{2}})^2+z^2)}^{3}\sqrt{x^{2}+y^{2}}}+7-7 =0$

So by Divergence theorem, we have:

$\int \int \int_R \nabla \cdot F(x,y,z) \,dx \,dy \,dz=\int \int_S F(x,y,z)\cdot \hat{n_{S}} \,dS_{S}+\int \int_{T_{1}} F(x,y,z)\cdot \hat{n_{T_{1}}} \,dS_{T_{1}}+\int \int_{T_{2}} F(x,y,z)\cdot \hat{n_{T_{2}}} \,dS_{T_{2}} (1)$

Where the surface S represent the region of the sphere limited by the plans, $T_{1}$ represents the intersection of the upper plan ( $z=3$ ) with the sphere and $T_{2}$ represents the intersection of the lower plan ( $z=-4$ ) with the sphere. But we doesn't can use the divergence theorem in this case because the flux, $F(x,y,z)$ , don't are limited in the region R. But, for the validity of the equation 1 we can make a improper surface integral around of the non limited region of the flux. So, we need add this for the right side of the equation 1:

$\lim_{\epsilon \to 0} \int \int_T F(x,y,z)\cdot \hat{n_{in}} \,dS.$

Where $\hat{n_{in}}$ is oriented for inside of the torus. And now we can calculate the flux over A, for the region $T_{1}$ , the $\hat{n_{T_{1}}}=(0,0,1)$ and for the region $T_{2}$ the $\hat{n_{T_{2}}}=(0,0,-1)$ , so we have:

$\int \int_{T_{1}} F(x,y,z)\cdot \hat{n_{T_{1}}} \,dS_{T_{1}}+\int \int_{T_{2}} F(x,y,z)\cdot \hat{n_{T_{2}}} \,dS_{T_{2}}= \int \int_{T_{1}} -7(3) \,dS + \int \int_{T_{2}} -(-7)(-4) \, dS$

$= -(21(\pi (5^{2}-3^{2}))+28(\pi (5^{2}-4^{2})))=-588\pi$

For find the flux of the torus, we need to use his parametric equation:

$\sigma(u,v)=((4-\epsilon cosv)cosu,(4-\epsilon cosv)sinu,\epsilon sinv)$

So, getting the opposite normal vector of this curve we have:

$\int \int_T F(x,y,z)\cdot \hat{n_{in}} = \int_0^{2 \pi} \int_0^{2 \pi} (\frac{-(4-\epsilon cosv)^{2}sinu cosu cosv \epsilon}{\sqrt{((4-\sqrt{x^{2}+y^{2}})^{2}+z^{2})}}) +(\frac{(4-\epsilon cosv)^{2}sinu cosu cosv \epsilon}{\sqrt{((4-\sqrt{x^{2}+y^{2}})^{2}+z^{2})}}$ $+7((4-\epsilon cosv)^{2}(sinu)^{2} cosv \epsilon)+ (-7(4-\epsilon cosv) \epsilon^{2} (sinv)^{2}) \,dv \,du$

$= \int_0^{2 \pi} \int_0^{2 \pi} +7((4-\epsilon cosv)^{2}(sinu)^{2} cosv \epsilon)+(-7(4-\epsilon cosv) \epsilon^{2} (sinv)^{2}) \,dv \,du=-112\pi^{2} \epsilon^{2}$

So,

$A=588\pi+\lim_{\epsilon \to 0} (112\pi^{2} \epsilon^{2})=588\pi$

And,

$a+b=588+112=700$

OFF: I think that is it. Sounds me cool make de k component of the flux be proportional a $\frac{z}{ \epsilon^2}$ , in this way may the flux of the torus be some number different of zero. But, it's hard make the divergent be zero in this case.