Curl (Vector Calculus)

The curl of $\vec{F}$ in $\mathbb{R}^3$ is itself a vector field, computed by formal cross product:

$\nabla \times \vec{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z},\ \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x},\ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right).$

Magnitude measures local rotation rate; direction is the rotation axis (right-hand rule).

A field with $\nabla \times \vec{F} = \vec{0}$ is irrotational — gradient (conservative) fields are always irrotational. Non-zero curl indicates local circulation.

Stokes' theorem equates surface integral of curl to line integral of $\vec{F}$ around the boundary. Used in EM (Maxwell-Faraday law), fluid dynamics (vorticity), aerodynamics.