I need to handle simple operation which needs some skill in tensor algebra. I have to take $\mathrm{rot}$ from $ (\vec u \cdot\nabla)\vec u $. I am not very good at tensors operations, but I know some basic rules.
So I have to do the following:
$$\mathrm{rot}((\vec u\cdot \nabla)\vec u) = \epsilon_{ijk} \nabla_j u_t \nabla_t u_k \tag{1}$$
Supposing that this is correct, now I have to derive:
$$\epsilon_{ijk} \nabla_j u_t \nabla_t u_k = \epsilon_{ijk} (\nabla_j u_t )*\nabla_t u_k + u_t * (\epsilon_{ijk} \nabla_j \nabla_t u_k),\tag{2}$$
where symbols * means only multiplication and parenthesis () ensure the order of action.
And I come to nothing, so I cannot finish my job here with what I wrote.
I tried also another approach:
$((\vec u\cdot \nabla)\vec u) = \nabla (u^2/2) - \vec u \times (\nabla \times u)$ and now:
$$\mathrm{rot}((\vec u \nabla)\vec u) = \mathrm{rot} (\nabla (u^2/2) - \vec u \times (\nabla \times u)) = 0 - \epsilon_{ijk} \nabla_j \epsilon_{klm} u_l \epsilon_{mzx} \nabla_z u_x \tag{3}$$
which in the end leads me to [2].
So am I wrong somewhere and could someone please help me with dealing
$$\mathrm{rot}((\vec u \cdot\nabla)\vec u) = ?$$
We first use the identity $$(\mathbf u \boldsymbol \cdot \boldsymbol \nabla)\mathbf u = \underbrace{\frac 12 \boldsymbol \nabla(\mathbf u \boldsymbol \cdot \mathbf u)}_{(1)} - \underbrace{\mathbf u \times (\boldsymbol \nabla \times \mathbf u)}_{(2)}.$$ When taking the curl, $(1)$ will vanish since the curl of a gradient is $\mathbf 0$. For $(2)$, \begin{align*} &-\boldsymbol \nabla \times \Big[\mathbf u\times (\boldsymbol \nabla \times \mathbf u)\Big] \\ ={}&-\varepsilon_{ijk}\partial_i [\varepsilon_{jpq} u_p [\boldsymbol \nabla \times \mathbf u]_q]_j \mathbf e_k \\ ={}& -\varepsilon_{jki} \partial_i \varepsilon_{jpq} u_p \varepsilon_{qmn} \partial_m u_n \mathbf e_k \\ ={}& -(\delta_{kp}\delta_{iq} - \delta_{kq}\delta_{ip}) \varepsilon_{qmn} \partial_i u_p\partial_mu_n \mathbf e_k \\ ={}& \varepsilon_{kmn} \partial_i u_i \partial_m u_n \mathbf e_k - \varepsilon_{imn} \partial_i u_k \partial_m u_n \mathbf e_k \\ ={}& (\partial_iu_i) \left(\varepsilon_{kmn} \partial_mu_n \mathbf e_k\right) - \left[(\varepsilon_{imn} \partial_mu_n)\partial_i\right]u_k \mathbf e_k \\ ={}& (\boldsymbol \nabla \boldsymbol \cdot \mathbf u)(\boldsymbol \nabla \times \mathbf u) - [(\boldsymbol \nabla \times \mathbf u) \boldsymbol \cdot \boldsymbol \nabla]\mathbf u. \end{align*}