The curl (also known as rotor) is a first order linear differential operatorMathworldPlanetmath which acts on vector fields in 3.

Intuitively, the curl of a vector field measures the extent to which a vector field differs from being the gradientMathworldPlanetmath 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 integralDlmfPlanetmath about a closed circuit.

Let F be a vector field in 3.

Pick an orthonormal basis {e1,e2,e3} and write F=F1e1+F2e2+F3e3. Then the curl of F, notated curlF or rotF or ×F, is given as follows:

curlF = [F3q2-F2q3]e1+[F1q3-F3q1]e2+

By applying the chain ruleMathworldPlanetmath, one can verify that one obtains the same answer irregardless of choice of basis, hence curl is well-defined as a functionMathworldPlanetmath 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


Where n is the outward unit normalMathworldPlanetmath vector to S.

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

curlF = 1h3h2[q2(h3F3)-q3(h2F2)]e1+1h3h1[q3(h1F1)-q1(h3F3)]e2+

for the arbitrary orthogonalMathworldPlanetmathPlanetmath curvilinear coordinate system (q1,q2,q3) having scale factors (h1,h2,h3). Note the scale factors are given by


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

Title curl
Canonical name Curl
Date of creation 2013-03-22 12:47:39
Last modified on 2013-03-22 12:47:39
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 17
Author rspuzio (6075)
Entry type Definition
Classification msc 53-01
Synonym rotor
Related topic IrrotationalField
Related topic FirstOrderOperatorsInRiemannianGeometry
Related topic AlternateCharacterizationOfCurl
Related topic ExampleOfLaminarField
Defines curl of a vector field