Einstein summation convention when a sum of terms is present

Question

I'm reading Landau / Liftshitz vol. 6 on fluid mechanics, and I encountered the expression (page 45, top):

$$\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}.$$

The expression above is referred to as a sum - hence the Einstein notation - and that got me thinking: Do I sum over both i and k, as

$$\sum_i \sum_k\left( \frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i} \right)$$

giving $9\cdot 2 = 18$ terms (in 3D)? I only ever encountered the summation convention in products before, but I guess both indices are repeated.

Thanks in advance

Edit:

I guess it calls for more context - thanks for ideas. It comes up when he's trying to deduce the form of the viscous stress tensor $\sigma_{ik}'$. Based on uniform rotation ($\vec{v} = \vec{\Omega} \times \vec{r}$) and linearity he concludes that only special linear combinations like the one i'm asking about can occur (so that when $\vec{v} = \vec{\Omega} \times \vec{r} \rightarrow \sigma_{ik}'=0$). He then concludes that the most general form of $\sigma_{ik}$ is:

$$\sigma_{ik}' = \eta \left( \frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i} - \frac{2}{3} \delta_{ik} \frac{\partial v_l}{\partial x_l} \right) + \zeta \delta_{ik} \frac{\partial v_l}{\partial x_l}$$

I couldn't really see how he concluded that last expression. I thought the clue might lie in the expression I first asked about, but it might be something else I'm missing.

Answer 1

Your formula for $\sigma_{ik}$ is $9$ equations, not one. If you set $k=i$, you sum over the now repeated $i$, obtaining the trace of $\sigma$. But if you sum the $9$ equations, you get something that isn't invariant under arbitrary linear coordinate transformations (the sum of entries of a matrix, even if square, is coordinate-dependent). This is why we don't sum over an index unless it's repeated per term ($A_i+B_i$ doesn't "repeat" $i$ in this sense).