$\mathsf{rot}U=\underset{V\to 0}{\lim }\frac{1}{V}{\displaystyle \underset{\text{S}}{∯}{e}_n\times U\mathrm{d}A}$

where e_ndA is the vector surface element oriented outwards and V is the volume

NOTE 1 In orthonormal Cartesian coordinates, the three coordinates of the rotation are:

${\displaystyle \frac{\partial U_z}{\partial y}-\frac{\partial U_y}{\partial z},\frac{\partial U_x}{\partial z}-\frac{\partial U_z}{\partial x},\frac{\partial U_y}{\partial x}-\frac{\partial U_x}{\partial y}}$

NOTE 2 The rotation of a polar vector is an axial vector and the rotation of an axial vector is a polar vector.

NOTE 3 The rotation of the vector field U is denoted by $\mathsf{rot}U$ or $\mathbf{\nabla }\times U$. In some English texts, the rotation is denoted by $\mathsf{curl}U$.