Problems with tensor notation I've got a question for the mathematically more educated for I am a humble engineer having a hard time:
$\kappa = \left( \delta_{ij}-n_in_j\right)\displaystyle\frac{\partial u_i}{\partial xj} - s_d\frac{\partial n_i}{\partial x_i}$
I don't see how this equation returns a scalar value, as it should for the second term obviously reassembles a divergence, which is a scalar and $\kappa$ is expected to be a scalar value as well. The first part of the formula is driving me nuts: for me it would translate into a product of two 3x3 tensors - which would result in a 3x3 tensor as well. Can anyone explain to me, what is the basic idea here? 
Hopefully I didn't miss a similar question, but searching for mathematical expressions is not straight forward at all.
Thanks for your time and effort!

If you believe the term $- s_d\frac{\partial n_i}{\partial x_i}$ is a scalar, then it needs to be a summation over $i$. If so, the other terms are likely to be summations as well, over both $i$ and $j$, and therefore you get scalars as well. It is hard to be more specific without knowing the origin of the formula, but it may be helpful to raise some of the indices to indicate that one is using the Einstein summation convention.
