Am not getting the right answer for $I = \int\limits_{S_\epsilon} \frac{x \,dy\,dz + y \,dx\,dz + z \,dx\,dy}{(x^2+y^2+z^2)^{\frac32}}$ I need to perform the following integration
$$I = \int\limits_{S_\epsilon} \frac{x \,dy\,dz + y \,dx\,dz + z \,dx\,dy}{(x^2+y^2+z^2)^{\frac32}},$$
where $S_{\epsilon}$ is a sphere of radius $\epsilon$ around the origin. So $(x^2+y^2+z^2)^{\frac12} = \epsilon$. By symmetry the integral reduces to
$$I=3\epsilon ^{-3}  \int_{S_\epsilon} z\,dx\,dy =3\epsilon ^{-3} \int_{B_\epsilon} \pm\sqrt{\epsilon ^2-x^2-y^2}\,dx\,dy .   $$
Here $B_{\epsilon}$ is a disc in the $xy$ plane of radius $\epsilon$. Now considering the plus or minus sign this would mean that $I = 0$ right? I think I made a mistake somewhere because the answer should be $4\pi$.
I found
$$ \int_{B_\epsilon} \sqrt{\epsilon ^2 -x^2 -y^2}\,dx\,dy = \frac{2\pi \epsilon^3}{3}. $$
I think I am missing something somwhere. Thanks in advance!
 A: As Jack D'aurizio suggested I used spherical coordinates and it was less painful than I thought. I posted this as an answer because I think that is better than editing the question and making it epicly long. I would still like to hear from someone how can spot the error in my initial method.
Let
$$ I = \int \limits_{S_{\epsilon}}   \frac{xdydz+ydzdx+zdxdy}{(x^2+y^2+z^2)^{\frac32}}.     $$
Here $S_{\epsilon}$ is a sphere of radius $\epsilon$ about the origin. To evaluate the integral we switch to spherical coordinates $(\theta, \phi)$. This becomes
$$ \begin{pmatrix}
x\\y\\z
\end{pmatrix}
=\epsilon\begin{pmatrix}
\cos\theta \sin \phi\\ \sin\theta \sin \phi \\ \cos \phi
\end{pmatrix}=f(\theta,\phi), \quad (\theta,\phi)\in [0,\pi)\times [0,2\pi). $$
Then we find that
$$ df = \begin{pmatrix}
-\epsilon \sin\theta\sin \phi & \epsilon\cos \theta \cos \phi\\
\epsilon \cos \theta \sin \phi & \epsilon \sin \theta \cos \phi\\
0 & -\epsilon \sin \phi
\end{pmatrix}.
$$
From this it follows that
$$\frac{\partial (x,y)}{\partial( \theta,\phi)}   =\epsilon^2\sin\phi\cos \phi,    $$
$$\frac{\partial (x,z)}{\partial( \theta,\phi)}   =\epsilon^2\sin^2 \phi \sin \theta,    $$
$$\frac{\partial (y,z)}{\partial( \theta,\phi)}   =\epsilon^2\sin^2 \phi \cos \theta.   $$
Noticing that the denominator in $I$ is simply equal to $\epsilon ^3$, performing the substitution yields
$$I = \int\limits_0^{2\pi} \int\limits_0^{\pi} [\sin^3\cos^2\theta +\sin^3\phi \cos^2\theta + \sin \phi \cos^2\phi] d\phi d\theta .$$
We can add the terms containing $\cos^2\theta$ and $\sin^2\theta$ and we are left with no terms involving $\theta$. This turns the integral into
$$I = 2\pi\int\limits_0^{\pi}      [      \sin^3\phi + \sin\phi \cos^2\phi] d\phi= 2\pi\int\limits_0^{\pi}      \sin\phi d\phi=-4\pi. $$
I am however left with a few questions still. I want to believe the workings above, however I think there might be one issue. When working out the absolute values of the Jacobians I simply left out the negative signs, however, the $\sin$ and $\cos$ terms might still be negative. In that case I would get some $|\sin|$ and $|\cos|$ terms in my integral in which case I need to split the integral in many parts. 
Secondly, I am still curious if someone can point out the mistake in the my initial method. Thanks
