In this document I cannot follow the expansion made by the author at equation 3.14. I reproduce it here for reference
$$\partial_j[\lambda \delta_{ij}\partial_k u_k+\mu(\partial_i u_j+\partial_j u_i)]$$
$$=\partial_i \lambda \partial_k u_k+\lambda \partial_i \partial_k u_k+\partial_j \mu (\partial_i u_j+\partial_j u_i)+\mu \partial_j \partial_iu_j+\mu \partial_j \partial_j u_i$$
Where $\partial_i = \frac{\partial}{\partial x_i}$. In the second expression, I can see there is a contraction going on namely for the first term $\partial_j\lambda\delta_{ij}\partial_ku_k = \partial_i\lambda\partial_ku_k$. I can also see that the second term from the first expression reoccurs as the third term in the second expression. However I have no clue where the other terms are coming from.
What are the intermediate steps necessary to arrive at the second expression? Could someone please provide a full expansion? Thanks
Let us first split the left side in two terms: $$ \partial_j[\lambda \delta_{ij}\partial_k u_k+\mu(\partial_i u_j+\partial_j u_i)] = \partial_j[\lambda \delta_{ij}\partial_k u_k] + \partial_j[\mu(\partial_i u_j+\partial_j u_i)] $$
For the first term we apply the product rule (noticing $\delta_{ij}$ is constant) and contract over $j$: $$ \partial_j[\lambda \delta_{ij}\partial_k u_k] = (\partial_j\lambda)\delta_{ij}\partial_ku_k + \lambda\delta_{ij}(\partial_j\partial_ku_k) = (\partial_i\lambda)\partial_ku_k + \lambda(\partial_i\partial_ku_k) $$
For the second term we do the same: $$ \partial_j[\mu(\partial_i u_j+\partial_j u_i)] = (\partial_j\mu)(\partial_i u_j+\partial_j u_i) + \mu \partial_j(\partial_i u_j+\partial_j u_i) \\ = (\partial_j\mu)(\partial_i u_j+\partial_j u_i) + \mu (\partial_j\partial_i u_j+\partial_j\partial_j u_i) $$
Adding the two results we reach the right hand side of you equation.