Curl

The Curl of a vector field $\mathbf{u}$ is…

$\nabla \times \mathbf{u}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial }{\partial x_1}& \frac{\partial }{\partial x_2}& \frac{\partial }{\partial x_3}\\ u_1& u_2& u_3\end{array}\right|=\left(\frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3},\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1},\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}\right)$.

Or, alternatively using the alternating tensor to represent the Curl, each component is then $[\nabla \times \mathbf{u}{]}_i={ϵ}_{ijk}{\nabla }_j{u}_k={ϵ}_{ijk}\frac{\partial u_k}{\partial x_j}$. Here, we can see the two dummy indices $j$ and $k$ indicating a double sum, and the free index $i$ indicating that the result is a vector quantity.

Vector in, vector out.