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  • ️Weisstein, Eric W.
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The curl of a vector field, denoted curl(F) or del xF (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of del xF is the limiting value of circulation per unit area. Written explicitly,

(del xF)·n^^=lim_(A->0)(∮_CF·ds)/A,

(1)

where the right side is a line integral around an infinitesimal region of area A that is allowed to shrink to zero via a limiting process and n^^ is the unit normal vector to this region. If del xF=0, then the field is said to be an irrotational field. The symbol del is variously known as "nabla" or "del."

The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in two of the four Maxwell equations,

where MKS units have been used here, E denotes the electric field, B is the magnetic field, mu_0 is a constant of proportionality known as the permeability of free space, J is the current density, and epsilon_0 is another constant of proportionality known as the permittivity of free space. Together with the two other of the Maxwell equations, these formulas describe virtually all classical and relativistic properties of electromagnetism.

In Cartesian coordinates, the curl is defined by

del xF=((partialF_z)/(partialy)-(partialF_y)/(partialz))x^^+((partialF_x)/(partialz)-(partialF_z)/(partialx))y^^+((partialF_y)/(partialx)-(partialF_x)/(partialy))z^^.

(4)

This provides the motivation behind the adoption of the symbol del x for the curl, since interpreting del as the gradient operator del =(partial/partialx,partial/partialy,partial/partialz), the "cross product" of the gradient operator with F is given by

del xF=|x^^ y^^ z^^; partial/(partialx) partial/(partialy) partial/(partialz); F_x F_y F_z|,

(5)

which is precisely equation (4). A somewhat more elegant formulation of the curl is given by the matrix operator equation

del xF=[0 -partial/(partialz) partial/(partialy); partial/(partialz) 0 -partial/(partialx); -partial/(partialy) partial/(partialx) 0]F

(6)

(Abbott 2002).

The curl can be similarly defined in arbitrary orthogonal curvilinear coordinates using

F=F_1u_1^^+F_2u_2^^+F_3u_3^^

(7)

and defining

h_i=|(partialr)/(partialu_i)|,

(8)

as

The curl can be generalized from a vector field to a tensor field as

(del xA)^alpha=epsilon^(alphamunu)A_(nu;mu),

(11)

where epsilon_(ijk) is the permutation tensor and ";" denotes a covariant derivative.

See also

Curl Theorem, Curvilinear Coordinates, Divergence, Gradient, Laplacian, Vector Derivative, Vector Laplacian

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References

Abbott, P. (Ed.). "Tricks of the Trade." Mathematica J. 8, 516-522, 2002.Arfken, G. "Curl, del x." §1.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 42-47, 1985.Kaplan, W. "The Curl of a Vector Field." §3.5 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 186-187, 1991.Morse, P. M. and Feshbach, H. "Curl." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 39-42, 1953.Schey, H. M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, 3rd ed. New York: W. W. Norton, 1997.

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Curl

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Weisstein, Eric W. "Curl." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Curl.html

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