How would I write a double dot product in index notation. For example, let $\vec{u} = [u_1,u_2], \vec{v}=[v_1,v_2]$, then
$ \nabla\vec{u}:\nabla\vec{v}=\nabla u_x\cdot\nabla v_x+\nabla u_y\cdot\nabla v_y= \frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1} + \frac{\partial u_1}{\partial x_2}\frac{\partial v_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}\frac{\partial v_2}{\partial x_1} + \frac{\partial u_2}{\partial x_2}\frac{\partial v_2}{\partial x_2} $
In index notation, I managed to get the first two terms in the right-most part of the equation:
$ \frac{\partial u_i}{\partial x_j}\frac{\partial v_i}{\partial x_j} = \frac{\partial u_1}{\partial x_1}\frac{\partial v_1}{\partial x_1} + \frac{\partial u_1}{\partial x_2}\frac{\partial v_1}{\partial x_2} $
where I have used implicit summation over the repeated index $j$... now that I look at it, it doesn't even seem correct, because there are two repeated indices...
Many thanks.
When you have $2$ repeated indices you sum over both both of them. So what you wrote (on the LHS) is actually correct. That is $$\frac{\partial u_i}{\partial x_j}\frac{\partial v_i}{\partial x_j}:= \sum_i \sum_j \frac{\partial u_i}{\partial x_j}\frac{\partial v_i}{\partial x_j}$$ which has all four terms.