I recently started using Einstein notation a lot from this paper AGARD-R-819. However, I was never formally introduced to it academically. I was wondering if someone here could help me validate my confusion. I have 3 questions.
1. $$\frac{\partial u_i}{\partial x_j} \; + \; \frac{\partial u_j}{\partial x_i}$$
From the equation, what impact does it have if you switch the $i$ & $j$. From my understanding, the answer is a scalar anyways so the order of the sum should not matter. I'm I correct?
2.The Kronecker delta basically tells us to sum the trace of a tensor.
$$\frac{\partial u_k}{\partial x_k} \: \delta_{ij}$$
So here the $k$ subscript is going to be k = 1, 2, 3. But why can't we write $j$ instead?
3.Similarly from 2,
$$\frac{\partial u_k}{\partial x_k} \: \delta_{ij} * \frac{\partial u_i}{\partial x_j}$$
Is it still going to be the sum of the trace for both derivative term?