In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point.
The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
If $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3 = \langle F_1(x,y,z),F_2(x,y,z),F_3(x,y,z)\rangle$, the curl of $\vec{F}$ can be computed as $$ \text{curl}(\vec{F}) = \left\langle \frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z},\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x},\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right \rangle $$Note that this is a vector, not a scalar (compare to the divergence). By a slight abuse of notation, if $\nabla$ is the gradient operator, we may write $$ \text{curl}(\vec{F}) = \nabla \times \vec{F}, $$where $\times$ is the cross product. Assuming equality of mixed partials, we have $\text{curl}(\nabla f)=\vec{0}$ for any scalar field $f$.
