I would like to show the following (we are in 2D, we apply Einstein's notation, and the norm is defined by $|n|=n_in_i$)
$$\mathbf{n} = n_x \mathbf{e}_x+n_y\mathbf{e}_y$$
$$(\nabla\cdot\mathbf{n})^{2}+(\mathbf{n}\times\nabla\times\mathbf{n})^{2}+(\mathbf{n}\cdot\nabla\times \mathbf{n})^{2}=|\nabla \mathbf{n}|^2:=\sum_{i=1}^2\sum_{j=1}^2(\partial_i n_j)^2$$
where $|\mathbf{n}|=1$, $\nabla\times\bf n=\partial_x n_y-\partial_y n_x$ and I didn't use Einstein's notation for the last term, in order to emphasise how it is to be intended.
what would be the quickest way possible, using Einstein's index notations or ...?