Cartesian Tensor Notation I have a problem in proving a problem, which is easy to do in basic multivariable calculus but I couldn't do it in tensor notation. Please help
$$\nabla.(r^n \vec{r})=(n+3)r^n$$
Here $r=\sqrt{x^2+y^2+z^2}$ is the magnitude of the position vector $\vec{r}=xi+yj+zk$.
I tried representing(i don't know if this is the right way either) $r$ like this $(x_ix_i)^{\frac{1}{2}}$ and $\vec{r}=x_j$ and $\nabla = {\frac{\partial}{\partial x_j}}$
Therefore, my representation is as follows:
$$\nabla.(r^n \vec{r})=(n+3)r^n$$
$$LHS={\frac{\partial}{\partial x_j}}((x_ix_i)^{\frac{n}{2}}x_j)$$ 
And I run into problems which I couldn't resolve like these terms appeared,
How do I turn this $x_mx_m(x_ix_i)^{\frac{n}{2}-1}$ into this  $(x_ix_i)^{\frac{n}{2}}$ 
So, I dont't know how to resolve these kind of problems. Please help. If my interpretation are wrong please tell me the right one.
 A: Firstly, while what you wrote is perfectly fine, it might be easier (pedagogically speaking) to only sum upper and lower indices, and make spatial variables contravariant (i.e. upper indices). Then you have to use the Kronecker delta, so that:
$$ x_ix_i \rightarrow \delta_{ij}x^ix^j $$
converts between our notations. 
For your last question about $x_mx_m(x_ix_i)^{n/2 - 1}$, I'd say:
$$ (\delta_{ij}x^ix^j)(\delta_{k\ell}x^kx^\ell)^{n/2 - 1} = (\delta_{k\ell}x^kx^\ell)(\delta_{k\ell}x^kx^\ell)^{n/2 - 1} = (\delta_{k\ell}x^kx^\ell)^{n/2} = r^n $$
since they are indeed dummy indices.
And then, with great explicitness (warning, this may not be stomachable to all viewers), I'd prove your identity via:
\begin{align}
\nabla\cdot(r^n\vec{r}) 
&= \partial_j [(\delta_{k\ell}x^kx^\ell)^{n/2}x^j] \\
&= x^j\partial_j (\delta_{k\ell}x^kx^\ell)^{n/2} + (\delta_{k\ell}x^kx^\ell)^{n/2}(\partial_jx^j) \\
&= x^j\partial_j (\delta_{k\ell}x^kx^\ell)^{n/2} + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\
&= x^j\frac{n}{2}(\delta_{k\ell}x^kx^\ell)^{n/2-1}[\delta_{k\ell}(x^k\delta^\ell_j + x^\ell\delta^k_j)] + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\ 
&= x^j\frac{n}{2}(\delta_{k\ell}x^kx^\ell)^{n/2-1}[2x^a\delta_{ja}] + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\ 
&= {n}(\delta_{k\ell}x^kx^\ell)^{n/2-1}[\delta_{ja}x^ax^j] + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\ 
&= {n}(\delta_{k\ell}x^kx^\ell)^{n/2-1}[\delta_{k\ell}x^kx^\ell] + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\ 
&= {n}(\delta_{k\ell}x^kx^\ell)^{n/2} + 3(\delta_{k\ell}x^kx^\ell)^{n/2} \\
&= ({n} + 3)(\delta_{k\ell}x^kx^\ell)^{n/2} \\ 
&= (n+3)r^n
\end{align}
