Compute $\iint_S \mathbf{F}\cdot d\mathbf{S}$ where $S$ is the surface that bounds the sphere $x^2+y^2+z^2=16$ and $\mathbf{F}=\langle z,y,x \rangle$ The problem is actually to verify the divergence theorem by computing both $\iiint_E \text{div } \mathbf{F\space} dV$, which was relatively easy to compute and gives $\frac{256\pi}{3}$. To find $\iint_S \mathbf{F}\cdot d\mathbf{S}$, I parametrized the surface with spherical coordinates: $\mathbf{r}=\langle 4\sin\phi\cos\theta,4\sin\phi\sin\theta,4\cos\phi\rangle$ with $0\leq\phi\leq\pi,0\leq\theta\leq2\pi$. Now, noting that $\iint_S\mathbf{F}\cdot d\mathbf{S}=\iint_S \mathbf{F}\cdot \mathbf{n}\space dS$ where $\mathbf{n}$ is the normal vector to the $S$, and since $S$ is a sphere, we have $\mathbf{n}=\langle\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi\rangle$, and $$\mathbf{F\cdot n}=4\sin\phi\cos\phi\cos\theta+4\sin^2\phi\sin\theta\cos\theta+4\sin\phi\cos\phi\cos\theta$$
But the integral of this is 0. 
Where am I going wrong?

On the surface of the sphere $\langle x,y,z \rangle = \langle \sin\phi\cos\theta, \sin\phi\sin\theta, \cos\phi \rangle$. 
Hence, $F = \langle z,y,x \rangle = \langle 4\cos\phi, 4\sin\phi\sin\theta,  4\sin\phi\cos\theta \rangle$. 
Also, $n = \frac{1}{4}\langle x,y,z \rangle = \langle \sin\phi\cos\theta, \sin\phi\sin\theta, \cos\phi \rangle$. 
Therefore, $F \cdot n = 4\cos\phi\sin\phi\cos\theta + 4\sin^2\phi\sin^2\theta + 4\cos\phi\sin\phi\cos\theta$. 
This shouldn't integrate to $0$. 
