Vector Index Notation - Simple Divergence Q has me really stumped? - seems to be a missing index? I'm teaching myself the very basics of index notation for vectors and the Einstein Summation Convention. I know that it's part of a larger field to do with tensors and such, but that's well above me, I'm just dealing with the very basics to do with vectors.
I'm stuck on exactly how to translate between a few simple examples done in 'normal notation' into index notation.
For example:

*

*Let the vector u = $<x^2,y^2,z^2>$. Then $\nabla\cdot $ u = $2x + 2y+2z$. Trying to do this in index notation however, I get stuck:

$\nabla\cdot$ u $= \partial_{i} u_{i} = \partial_{i}(x_{j}^2) = 2x_{j}\partial_{i}(x_{j})=2x_{i}\delta_{ij}$
Now, here's where I'm stuck. Assuming I've been correct up til now (which I most probably haven't been), shouldn't $2x_{i}\delta_{ij} = 2x_{j}$ for the rule where summing over $_{i}$ with the kronecker in the term will mean that $x_{i}\delta_{ij} = x_{j} \cdot(1)$ only when i = j?
But, if this is so, then $\nabla\cdot$ u $=2x_{j}$ - but that doesn't make sense to me, because j is now a free variable, and we need to sum (thus a double index?) over j = 1, 2 and 3?
What's the rule for sorting this?

*

*Say again I wanted to calculate the divergence, or perhaps the gradient, etc, but for a function that can't 'neatly' be written in index form - how then do I write it in index form?

E.g. - Let v = $<x, x+y, 10> $. This isn't a vector that, to my knowledge, can easily be written in index form, such as u from example 1, i.e. u $= x_{i}^2$. If then I wanted to do $\nabla \cdot $ v, would I just do $\nabla \cdot $ v $= \partial_{i}u_{i} = \partial_{i}( <x, x+y, 10>) $, and go from there? But even then, I've no idea how to handle that in index notation?
Would I instead just have to 'abandon' trying to write it all in one closed form, and go straight to writing out the sum 'by hand'?
$\nabla \cdot $ v $= \partial_{i}u_{i} = \partial_{x}(x) + \partial_{y}(y) + \partial_{z}(z)$
What's the step-by-step on this, and the overall idea? Thanks!

Thanks so much for your time and help! I know this is super basic, but in teaching myself I'm really struggling to find good resources that go through the basics.
With that in mind, does anyone have such recommendations - of question books etc (with solutions, of course), that start right at the beginning and go over all the very basics of translating between 'normal' and index notation, so I can learn the nuances?
Many thanks for your help!
 A: Einstein conceived his sum convention as a short notation for the application of multilinear maps on tangent vectors. In your example, the numbers $u_1,u_2$ and $u_3$ are the component functions of a vector: $u_1$ is the first component, $u_2$ is the second and $u_3$ is the third.
In contrast, $x_1,x_2$ and $x_3$ are not the components of any vector: they are the coordinate functions of the space. It is important to distinguish between the two: coordinate functions transform with diffeomorphisms, such as
$$\begin{align*}
x_1&=y_1\sin y_2\cos y_3 \\
x_2&=y_1\sin y_2\sin y_3 \\
x_3&=y_1\cos y_2 \\
\end{align*}$$
(the $x_i$ may be thought of as cartesian coordinates and the $y_i$ as cylindrical coordinates).
In contrast, vector components transform with the derivative of the diffeomorphism, which is a linear transformation:
$$\begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}
=
\begin{pmatrix}
 \sin y_2\cos y_3 & \sin y_2\sin y_3 & \cos y_2 \\
 \cos y_2\cos y_3 & \cos y_2\sin y_3 & -\sin y_2 \\
 -\sin y_3 & \cos y_3 & 0
\end{pmatrix}
\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}
$$
That is why you got into something tricky: you are trying to apply the summation convention to indices that do not indicate the components of a vector, but the coordinates of your space. The solution is just not applying this convention and keep the sigma signs as follows
$$\sum_i\partial_i(x_i^2) = \sum_i2x_i\partial_ix_i=2\sum_ix_i\delta_{ii}=2(x_1\delta_{11}+x_2\delta_{22}+x_3\delta_{33})=2(x_1+x_2+x_3)$$
as expected.

Now, having said that,
sometimes one can force the sum convention to work. As I mentioned above, it was intended for writing multilinear maps, so you need to write everything in terms of multilinear maps. In this case you can write $x_i^2$ as $x_i^2=x_j\delta_{ijk}x_k$, where
$$\delta_{ijk}=
\begin{cases}
1 & i=j=k \\
0 & \text{else}
\end{cases}$$
are the components of a third order tensor: for each $i$ we get a bilinear form. Now, with this in mind you can proceed normally
$\partial_iu_i=\delta_{ijk}\partial_i(x_jx_k) = \delta_{ijk}(\delta_{ij}x_k+x_j\delta_{ik}) = \delta_{iik}x_k+\delta_{iji}x_j$
and at this point we need to know: what is $\delta_{iik}$? For each $k$ we have
$$\delta_{iik}=\delta_{11k}+\delta_{22k}+\delta_{33k}=
1$$
so $\delta_{iik}x_k=1x_1+1x_2+1x_3=\sum_ix_i$. Similarly $\delta_{iji}x_j=\sum_ix_i$ and hence $\partial_iu_i=\sum_ix_i+\sum_ix_i=2\sum_ix_i$, as you wanted.
