# Einstein summation convention when a sum of terms is present

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.

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.

• I haven’t got L&L but is is very unusual to assume non-repeated indices are summed over. Usually one adds the summation symbol explicitly. Feb 23 at 16:18
• Well, it literally is a sum of two terms. Are you sure L&L mean something more complicated? Feb 23 at 16:53
• Can you give more context? Often you can infer from the context whether that object should still have indices or not.
– noah
Feb 23 at 16:57
• Einstein summation only occurs when an index appears twice in the same term. Feb 23 at 17:57
• It's the Strain Rate Tensor. en.wikipedia.org/wiki/…
– user288901
Feb 24 at 7:34

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).