# Math Help - divergence theorem problem

1. ## divergence theorem problem

F= (x^3+2xy^2,y^3+2yz^2,z^3+2zx^2) is a vector field.

fine integral( F.dS) on a sphere of radius R centered at the origin.

I am not sure about what dS is. I think it should be a unit vector perpendicular to the surface u r calculating, is it right?
I let n = ( 0, 0, 1) and try to integrate, but there are z and x , how to operate this?

2. Let's say the sphere has radius a.

$(x^{3}+2xy^{2})i+(y^{3}+2yz^{2})j+(z^{3}+2zx^{2})k$

sphere $x^{2}+y^{2}+z^{2}=a^{2}$

$\frac{\partial}{{\partial}x}(x^{3}+2xy^{2})=x^{3}+ 2xy^{2}$

$\frac{\partial}{{\partial}y}(y^{3}+2yz^{2})=3y^{2} +2z^{2}$

$\frac{\partial}{{\partial}z}(z^{3}+2zx^{2})=3z^{2} +2x^{2}$

Adding and factoring yields $5(x^{2}+y^{2}+z^{2})=5{\rho}^{2}$

$5\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{a}{\rho}^{ 4}sin{\phi}d{\rho}d{\phi}d{\theta}$