I would like to know how can I write $ ||\vec{a} \times(\nabla \times \vec{a})||^2 $ and $(\vec{a} \cdot (\nabla \times \vec{a}))^2$ in index notation if $\vec{a}=(a_1,a_2,a_3)$
Thank you for reading/replying
EDIT: found the second one: $(\vec{a} \cdot (\nabla \times \vec{a}))^2 = a_ia_ja_{k,i}a_{k,j}$ The first one can also be written as $ ||\vec{a} \times(\nabla \times \vec{a})||^2 = (a_ie_{ijk}a_{k,j})^2 $ but if one finds a better expression let me know!
To do this I would use the Levi-Civita symbol and its properties in 3 dimensions.
(from Wikipedia:)
Definition:
\begin{equation} \varepsilon_{ijk}= \left\{ \begin{array}{l} +1 \quad \text{if} \quad (i,j,k)\ \text{is}\ (1,2,3),(3,1,2)\ \text{or}\ (2,3,1)\\ -1 \quad \text{if} \quad (i,j,k)\ \text{is}\ (1,3,2),(3,2,1)\ \text{or}\ (2,1,3)\\ \ \ \ 0\quad \text{if} \quad i=j\ \text{or}\ j=k\ \text{or}\ k=i \end{array} \right. \end{equation}
Vector product:
\begin{equation} \vec a\times \vec b=\sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}\vec e_ia^jb^k \end{equation}
Component of a vector product:
\begin{equation} (\vec a\times \vec b)_i=\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}a^jb^k \end{equation}
Spatproduct
\begin{equation} \vec a\cdot(\vec b\times\vec c)=\vec a\times \vec b=\sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}a^ib^jc^k \end{equation}
Useful properties:
\begin{equation} \sum_{i=1}^3\varepsilon_{ijk}\varepsilon^{imn}=\delta_j^{\ m}\delta_k^{\ n}-\delta_j^{\ n}\delta_k^{\ m} \end{equation}
\begin{equation} \sum_{m=1}^3\sum_{n=1}^3\varepsilon_{jmn}\varepsilon^{imn}=2\delta_{\ j}^{i} \end{equation}
\begin{equation} \sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}\varepsilon^{ijk}=6 \end{equation}