How does one deal with modulus in index notation?

Question

I'm new to using index notation and the Einstein Summation Convention in calculus.

Where $\underline{a}$ and $\underline{r}$ are both vectors in $\mathbb{R}^{3}$ and by $_i$ I mean the $ i^{th}$ component:

\begin{align} \nabla(|\underline{r}-\underline{a}|) & = \nabla(|x_i - a_i|)\\ & = \nabla[((x_i - a_i)^{1/2})^{1/2}] \end{align}

On this final line the rules of algebra do not follow. I have included the $\nabla $ as I wish to move further with the notation, arriving at a line $[\nabla(|\underline{r}-\underline{a}|)]_i = ...$

So essentially my question is: how does one deal with modulus in index notation?

I am very new to this notation so many apologies if this question is unclear!

Answer 1

$\nabla(|\underline{r}-\underline{a}|) = \partial_i[(r_j -a_j)(r_j -a_j)]^{1/2}$ Summation over $j$ is implied by the convention and $\partial_i$ implies all three components. But really one wouldn't use index notation here as it is simpler with components!