The curl (also known as rotor) is a first order linear differential operator which acts on vector fields in $\mathbb{R}^{3}$ .

Intuitively, the curl of a vector field measures the extent to which a vector field differs from being the gradient of a scalar field. The name "curl" comes from the fact that vector fields at a point with a non-zero curl can be seen as somehow "swirling around" said point. A mathematically precise formulation of this notion can be obtained in the form of the definition of curl as limit of an integral about a closed circuit.

Let $F$ be a vector field in $\mathbb{R}^{3}$ .

Pick an orthonormal basis $\{\vec{e_{1}},\vec{e_{2}},\vec{e_{3}}\}$ and write $\vec{F}=F^{1}\vec{e_{1}}+F^{2}\vec{e_{2}}+F^{3}\vec{e_{3}}$ . Then the curl of $F$ , notated $\operatorname{curl}\vec{F}$ or $\operatorname{rot}\vec{F}$ or $\vec{\nabla}\times\vec{F}$ , is given as follows:

\begin{eqnarray*} \operatorname{curl}\vec{F} & = & \left[\frac{\partial F^{3}}{\partial q^{2}}-\frac{\partial F^{2}}{\partial q^{3}}\right]\vec{e_{1}}+\left[\frac{\partial F^{1}}{\partial q^{3}}-\frac{\partial F^{3}}{\partial q^{1}}\right]\vec{e_{2}} + \\ & \; & \left[\frac{\partial F^{2}}{\partial q^{1}}-\frac{\partial F^{1}}{\partial q^{2}}\right]\vec{e_{3}} \end{eqnarray*} By applying the chain rule, one can verify that one obtains the same answer irregardless of choice of basis, hence curl is well-defined as a function of vector fields. Another way of coming to the same conclusion is to exhibit an expression for the curl of a vector field which does not require the choice of a basis. One such expression is as follows: Let $V$ be the volume of a closed surface $S$ enclosing the point $p$ . Then one has $$ \operatorname{curl}\vec{F}(p)=\lim_{V\to 0}\frac{1}{V}\int\!\!\int_{S}\vec{n}\times\vec{F}dS$$

Where $n$ is the outward unit normal vector to $S$ .

Curl is easily computed in an arbitrary orthogonal coordinate system by using the appropriate scale factors. That is

\begin{eqnarray*} \operatorname{curl}\vec{F} & = & \frac{1}{h_{3}h_{2}}\left[\frac{\partial}{\partial q^{2}}\left(h_{3}F^{3}\right)-\frac{\partial}{\partial q^{3}}\left(h_{2}F^{2}\right)\right]\vec{e_{1}}+\frac{1}{h_{3}h_{1}}\left[\frac{\partial}{\partial q^{3}}\left(h_{1}F^{1}\right)-\frac{\partial}{\partial q^{1}}\left(h_{3}F^{3}\right)\right]\vec{e_{2}} + \\ & \; & \frac{1}{h_{1}h_{2}}\left[\frac{\partial}{\partial q^{1}}\left(h_{2}F^{2}\right)-\frac{\partial}{\partial q^{2}}\left(h_{1}F^{1}\right)\right]\vec{e_{3}} \end{eqnarray*} for the arbitrary orthogonal curvilinear coordinate system $(q^{1},q^{2},q^{3})$ having scale factors $(h_{1},h_{2},h_{3})$ . Note the scale factors are given by $$ h_{i}=\left(\frac{d}{dx_{i}}\right)\left(\frac{d}{dx_{i}}\right)\;\ni\; i\in \{1,2,3\}.$$

Non-orthogonal systems are more easily handled with tensor analysis or exterior calculus. $$ (\operatorname{curl}\vec{F})^i = \epsilon^{ijk} \nabla_j F_k$$ $$ \operatorname{curl}\vec{F} = * d (F_1 dx^1 + F_2 dx^2 + F_3 dx^3)$$