The curl (also known as rotor) is a first order linear differential operator which acts on vector fields in .

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 be a vector field in .

Pick an orthonormal basis and write . Then the curl of , notated or or , is given as follows:

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 be the volume of a closed surface enclosing the point . Then one has

Where is the outward unit normal vector to .

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

for the arbitrary orthogonal curvilinear coordinate system having scale factors . Note the scale factors are given by

Non-orthogonal systems are more easily handled with tensor analysis or exterior calculus.