Definition
The curl of a vector field F, denoted by curl F or ∇ × F, at a point is defined in terms of its projection onto various lines through the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
As such, the curl operator maps C1 functions from R3 to R3 to C0 functions from R3 to R3.
Implicitly, curl is defined by:
where is a line integral along the boundary of the area in question, and |A| is the magnitude of the area. If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane (see caption at right), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R3 (right-hand rule).
The above formula means that the curl of a vector field is defined as the infinitesimal area density of the circulation of that field. To this definition fit naturally
- the Kelvin-Stokes theorem, as a global formula corresponding to the definition, and
- the following "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in cartesian coordinates, spherical, cylindrical, or even elliptical or parabolical coordinates:
If (x1, x2, x3) are the Cartesian coordinates and (u1,u2,u3) are the orthogonal coordinates, then
is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.
Read more about this topic: Curl (mathematics)
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lensif we are unaware that women even have a historywe live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on life (based on wording in the First Edition, 1935)