I'm having some trouble using index notation to prove the identity
$$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$
The closest I can get is by expanding the first term on the RHS, which gives
$$\mathbf{u \times (\nabla \times u)} = 2u_j \partial x_i u_j – u_j\partial x_i u_i – u_i \partial x_j u_j$$
but I don't see what to do from here (if what I've done so far is correct).
Any help will be appreciated!
EDIT
The comments so far are all a bit dubious about my expression for the first term on the RHS, here's my work:
$$\mathbf{u \times (\nabla \times u)} = \epsilon_{ijk}u_j\epsilon_{klm}\partial x_lu_m$$
now I move the second Levi-Civata symbol to the left and use the identity GFR posted to get
$$\epsilon_{ijk}u_j\epsilon_{klm}\partial x_lu_m = (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})u_j\partial x_lu_m$$
expanding this gives
$$\mathbf{u \times (\nabla \times u)} = u_j\partial x_i u_j – u_j\partial x_j u_i$$
This next step i'm not sure about, I move the derivatives to the left of each term
$$u_j\partial x_i u_j – u_j\partial x_j u_i = \partial x_iu_ju_j – \partial x_j u_ju_i$$
then the product rule gives my original equation for $\mathbf{u \times (\nabla \times u)}$
Best Answer
So, your first step is indeed correct, but the one you are not sure about is wrong. You are probably confused by your own notation, I prefer to write $\partial_i=\frac{\partial}{\partial x_i}$ instead of $\partial x_i$. And remember that repeated indices are being summed over, so e.g. $u_ju_j=\mathbf{u}\cdot\mathbf{u}$. Given that, you have \begin{equation} u_j\partial _i u_j-u_j\partial_j u_i =(1/2)\partial_i(u_j u_j)-(\mathbf{u}\cdot \nabla)u_i =(1/2) \partial_i (\mathbf{u}\cdot\mathbf{u})-(\mathbf{u}\cdot \nabla)u_i, \end{equation} which (taking into account your earlier work) is the $i$-th component of the equation \begin{equation} \mathbf{u}\times(\nabla\times \mathbf{u})=\frac{1}{2}\nabla(\mathbf{u}\cdot\mathbf{u})-(\mathbf{u}\cdot \nabla)\mathbf{u} \end{equation}