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$$\begin{array}{c} \begin{matrix} \delta_{ij}=\begin{cases}1&i=j\\0&i\ne j\end{cases}& \epsilon_{ijk}=\begin{cases}1&\text{cyclic}\\-1&\text{anti-cyclic}\\0&\text{repeated}\end{cases} \end{matrix}\\ \begin{matrix} (a\times b)_k=\epsilon_{ijk}a_ib_j& a\times b=\begin{Bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{Bmatrix}& \tilde a=\begin{bmatrix}0&-a_3&a_2\\a_3&0&-a_1\\-a_2&a_1&0\end{bmatrix} \end{matrix}\\ a\cdot b=a_ib_i\\ a_{\parallel b}=\frac{a\cdot b}{b\cdot b}b\\ a_{\perp b}=a-a_{\parallel b}\\ \cos\theta=\frac{a\cdot b}{|a||b|}\\ \nabla=\partial_i\\ \nabla p=p_i\\ \nabla\vec v=u_{i,j}\\ \nabla\cdot\vec v=u_{i,i}\\ u\otimes u=u_iu_j\\ \nabla (u\otimes u)=\nabla uu=u_{i,j}u_j\\ D\rho=\rho_{,t}+u_i\rho_{,i}\\ \end{array}$$