Integrate outward unit vector over the surface of a sphere I'm trying to understand an integration over the surface of a sphere that is used in one of the articles I'm reading. I don't know why I can't understand it as it seems to be a pretty straightforward integration and I have been used to more complex math but anyway.
Let $\textbf{r} = \textbf{x - x'}$, $r = |\textbf{r}|$ and $\textbf{n} = \frac{\textbf{r}}{r}$ where $|.|$ is the euclidian norm. The goal is to calculate the following integral for $i, j \in \{1,2,3\}$ :
$$I = \int_{A(r)} n_{i}n_{j}\text{d}A$$
Where $A(r)$ denotes a spherical surface of radius $r$.
The way I've gone about it is to say that for a spherical surface of radius $r$ we have 
$$\text{d}A = r\sin(\phi)\text{d}\phi\text{d}\theta$$
with $(\phi, \theta) \in [0,\pi]\times[0,2\pi]$ and since $r_{i}$ or $r_{j}$ are not dependent of the angles we should have 
$$I = 4\pi r_{i}r_{j}$$
However the article I'm reading has the result 
$$I = \frac{4\pi r^{2}}{3} \delta_{ij}$$
I've been thinking about it but can't seem to find my mistake.
Thanks for your help kind stranger :)
 A: Given your choice of parameters, you seem to be parametrizing $A(r)$ via
\begin{equation}
(\phi,\theta) \mapsto (r\cos\theta\sin\phi, r\sin\theta\sin\phi, r\cos\phi).
\end{equation}
This allows us to parametrize the unit normal vector $\mathbf{n}$:
\begin{equation}
\mathbf{n}(\phi,\theta) = (\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi).
\end{equation}
From this we have
\begin{align*}
n_1n_{1} &= \cos^2\theta\sin^2\phi\\
n_1n_2 &= \sin\theta\cos\theta\sin^2\phi\\
n_1n_3 &= \cos\theta\sin\phi\cos\phi\\
n_2n_2 &= \sin^2\theta\sin^2\phi\\
n_2n_3 &= \sin\theta\sin\phi\cos\phi\\
n_3n_3 &= \cos^2\phi.
\end{align*}
We then have
\begin{equation}
I_{ij} = \int_0^{2\pi}\int_0^\pi n_in_j r^2\sin\phi d\phi d\theta.
\end{equation}
(Notice that the integrating factor should be $dA=r^2\sin\phi d\phi d\theta$.  You can get this by computing the relevant Jacobian determinant for your parametrization.)  We can then directly compute the six integrals.  For instance,
\begin{align*}
I_{11} &= \int_0^{2\pi}\int_0^\pi r^2\cos^2\theta\sin^3\phi d\phi d\theta = r^2\left(\int_0^{2\pi}\cos^2\theta d\theta\right)\left(\int_0^\pi \sin^3\phi d\phi\right)\\ &= r^2\cdot \pi\cdot \frac{4}{3} = \frac{4\pi r^2}{3}\delta_{11}.
\end{align*}
The other integrals will work out similarly.  The real problem seems to have been the integrating factor.  Notice that if $i\neq j$, then $n_in_j$ has an odd-power $\theta$ term, and this will kill the integral.
