I am following a text on turbulence by Frisch. Using the summation convention, he writes the Navier-Stokes equations as $$\partial_t v_i + v_j\partial_j v_i = - \partial_i p + \nu \partial_{jj}v_i\\ \partial_iv_i = 0.$$ where $\nu$ is a constant. He claims that taking the divergence above results in a Poisson equation.
Taking the divergence results in : $$\partial_i \partial_t v_i + \partial_i v_j\partial_j v_i = -\partial_i \partial_i p + \partial_i \nu \partial_{jj}v_i.$$ Assuming we may interchange partial derivatives, we have $$\partial_t \partial_i v_i + \partial_i v_j\partial_j v_i = -\partial_i \partial_i p + \nu \partial_{jj}\partial_iv_i.$$ By the divergence free condition the first and last terms vanish, leaving $$\partial_i v_j\partial_j v_i = -\partial_i \partial_i p \\ \implies\partial_{ij}(v_iv_j) = -\partial_{ii}p.$$
The question I have now is what justifies interchanging the partial derivatives as we did?
Also as an aside, I am still grasping the summation notation. How can we think of the term $\partial_{ij}(v_iv_j)$? Since the indices are not repeated adjacently, is it not summed?