So far I have: $$||\nabla\times\underline{u}||^2 = \left[(\nabla\times\underline{u})\right]_i\left[(\nabla\times\underline{u})\right]_k =\left[\varepsilon_{ijk}\frac{\partial u_k}{\partial x_j}\right]\left[\varepsilon_{kmn}\frac{\partial u_n}{\partial x_m}\right]$$ $$=\varepsilon_{kij}\varepsilon_{kmn}\frac{\partial u_k}{\partial x_j}\frac{\partial u_n}{\partial x_m} = \left[\delta_{im}\delta_{jn}-\delta_{in}\delta_{jm}\right]\frac{\partial u_k}{\partial x_j}\frac{\partial u_n}{\partial x_m}$$ $$= \frac{\partial u_k}{\partial x_j}\frac{\partial u_j}{\partial x_i} - \frac{\partial u_k}{\partial x_j}\frac{\partial u_i}{\partial x_j}$$ Here is where I don't understand how I am to get the final result. I struggle on this a lot so if someone could show me that would be great.
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In general, $|\vec a |^2 = \vec a \cdot \vec a = a_i a_i$. At the beginning, you said $a_ia_k$. There is your problem. – Dog_69 Feb 02 '18 at 17:21
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Anyway, I don't understand what is $\nabla u$, because you wan to compute its norm (so I think it is a vector) and its trace (so it is a matrix). How? – Dog_69 Feb 02 '18 at 17:25
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$\nabla u$ is the div of u. But if I use $a_i a_i$ then I don't get the right result with the kroneker delta. – MRT Feb 03 '18 at 10:26
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Yes. What really happens is your calculus is wrong. You can't use $k$ index in the first $\epsilon_{kij}$. Do you know the formula $mistake + mistake = hit$? On the other hand, If $\nabla u$ is the divergence of $u$, then it is a scalar. How do you define the trace and the norm of a scalar? I think your post has a lot of mistakes. – Dog_69 Feb 03 '18 at 16:05
1 Answers
By $\boldsymbol \nabla \mathbf u$, I assume you mean the Jacobian matrix of $\mathbf u$, with components $$J_{ij} = \partial_j u_i = \frac{\partial u_i}{\partial x_j}.$$ Then, \begin{align*} \lVert \boldsymbol \nabla \times \mathbf u \rVert^2 &= [\boldsymbol \nabla \times \mathbf u]_i[\boldsymbol \nabla \times \mathbf u]_i \\ &= \varepsilon_{ijk}\partial_ju_k \varepsilon_{ipq}\partial_pu_q \\ &= (\delta_{jp}\delta_{kq} - \delta_{jq}\delta_{kp})\partial_ju_k\partial_p u_q \\ &= \partial_ju_k \partial_j u_k - \partial_j u_k \partial_k u_j \\ &= J_{kj}J_{kj} - J_{jk} J_{kj} \\ &= \left[\mathbf J \mathbf J^T\right]_{kk} - \left[\mathbf J^2\right]_{jj} \\ &= \operatorname{Tr}\left(\mathbf J \mathbf J^T\right) - \operatorname{Tr} \left(\mathbf J^2\right). \end{align*} The original notation in your question is confusing and ambiguous.
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