Curl (mathematics)
In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in threedimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation.[1] The curl of a field is formally defined as the circulation density at each point of the field.
A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.
The alternative terminology rotation or rotational and alternative notations rot F or the cross product with the del (nabla) operator ∇ × F are sometimes used for curl F. The ISO/IEC 800002 standard recommends the use of the rot notation in boldface as opposed to the curl notation.[2]
Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; when expressed via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The unfortunate circumstance is similar to that attending the 3dimensional cross product, and indeed the connection is reflected in the notation ∇× for the curl.
The name "curl" was first suggested by James Clerk Maxwell in 1871[3] but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.[4][5]
Definition
The curl of a vector field F, denoted by curl F, or ∇ × F, or rot 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 divided by the area enclosed, as the path of integration is contracted around the point.
The curl operator maps continuously differentiable functions f : R^{3} → R^{3} to continuous functions g : R^{3} → R^{3}, and in particular, it maps C^{k} functions in R^{3} to C^{k−1} functions in R^{3}.
Implicitly, curl is defined at a point p as[6][7]
where the line integral is calculated along the boundary C of the area A in question, A being the magnitude of the area. This equation defines the projection of the curl of F onto . The infinitesimal surfaces bounded by C have as their normal. C is oriented via the righthand 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 parabolic coordinates:
The equation for each component (curl F)_{k} can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices).
If (x_{1}, x_{2}, x_{3}) are the Cartesian coordinates and (u_{1}, u_{2}, u_{3}) are the orthogonal coordinates, then
is the length of the coordinate vector corresponding to u_{i}. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.
Intuitive interpretation
Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.[8]
The curl of the vector at any point is given by the rotation of an infinitesimal area in the xyplane (for zaxis component of the curl), zxplane (for yaxis component of the curl) and yzplane (for xaxis component of the curl vector). This can be clearly seen in the examples below.
Usage
In practice, the above definition is rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.
The notation ∇ × F has its origins in the similarities to the 3dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra.
Expanded in 3dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations),∇ × F is, for F composed of [F_{x}, F_{y}, F_{z}] (where the subscripts indicate the components of the vector, not partial derivatives):
where i, j, and k are the unit vectors for the x, y, and zaxes, respectively. This expands as follows:[9]^{:43}
Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.
In a general coordinate system, the curl is given by[1]
where ε denotes the LeviCivita tensor, ∇ the covariant derivative, is the Jacobian and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative:
where R_{k} are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as:
Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented threedimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.
Examples
Example 1
The vector field
can be decomposed as
Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed.
Calculating the curl:
The resulting vector field describing the curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the righthand rule using a righthanded coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.
Example 2
For the vector field
the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the righthand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3, the object would rotate counterclockwise and the righthand rule would result in a positive z direction.
Calculating the curl:
The curl points in the negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0.
Descriptive examples
 In a vector field describing the linear velocities of each part of a rotating disk, the curl has the same value at all points.
 Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and rate of change of the electric field.
Identities
In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be
Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field:
where ∇_{F} is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).
Another example is the curl of a curl of a vector field. It can be shown that in general coordinates
and this identity defines the vector Laplacian of F, symbolized as ∇^{2}F.
The curl of the gradient of any scalar field φ is always the zero vector field
which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives.
If φ is a scalar valued function and F is a vector field, then
Generalizations
The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0forms, 1forms, and 2forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2vectors) in 3 dimensions with the special orthogonal Lie algebra (3) of infinitesimal rotations (in coordinates, skewsymmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1vectors (equivalently, 2vectors) and (3), these all being 3dimensional spaces.
Differential forms
In 3 dimensions, a differential 0form is simply a function f(x, y, z); a differential 1form is the following expression:
a differential 2form is the formal sum:
and a differential 3form is defined by a single term:
(Here the acoefficients are real functions; the "wedge products", e.g. dx ∧ dy, can be interpreted as some kind of oriented area elements, dx ∧ dy = −dy ∧ dx, etc.)
The exterior derivative of a kform in R^{3} is defined as the (k + 1)form from above—and in R^{n} if, e.g.,
then the exterior derivative d leads to
The exterior derivative of a 1form is therefore a 2form, and that of a 2form is a 3form. On the other hand, because of the interchangeability of mixed derivatives, e.g. because of
the twofold application of the exterior derivative leads to 0.
Thus, denoting the space of kforms by Ω^{k}(R^{3}) and the exterior derivative by d one gets a sequence:
Here Ω^{k}(R^{n}) is the space of sections of the exterior algebra Λ^{k}(R^{n}) vector bundle over R^{n}, whose dimension is the binomial coefficient (^{n}
_{k}); note that Ω^{k}(R^{3}) = 0 for k > 3 or k < 0. Writing only dimensions, one obtains a row of Pascal's triangle:
 0 → 1 → 3 → 3 → 1 → 0;
the 1dimensional fibers correspond to scalar fields, and the 3dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div.
Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudoRiemannian manifold, kforms can be identified with kvector fields (kforms are kcovector fields, and a pseudoRiemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between kvectors and (n − k)vectors; in particular on (the tangent space of) an oriented pseudoRiemannian manifold. Thus on an oriented pseudoRiemannian manifold, one can interchange kforms, kvector fields, (n − k)forms, and (n − k)vector fields; this is known as Hodge duality. Concretely, on R^{3} this is given by:
 1forms and 1vector fields: the 1form a_{x} dx + a_{y} dy + a_{z} dz corresponds to the vector field (a_{x}, a_{y}, a_{z}).
 1forms and 2forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ∧ dy. Thus the form a_{x} dx + a_{y} dy + a_{z} dz corresponds to the "dual form" a_{z} dx ∧ dy + a_{y} dz ∧ dx + a_{x} dy ∧ dz.
Thus, identifying 0forms and 3forms with scalar fields, and 1forms and 2forms with vector fields:
 grad takes a scalar field (0form) to a vector field (1form);
 curl takes a vector field (1form) to a pseudovector field (2form);
 div takes a pseudovector field (2form) to a pseudoscalar field (3form)
On the other hand, the fact that d^{2} = 0 corresponds to the identities
for any scalar field f, and
for any vector field v.
Grad and div generalize to all oriented pseudoRiemannian manifolds, with the same geometric interpretation, because the spaces of 0forms and nforms at each point are always 1dimensional and can be identified with scalar fields, while the spaces of 1forms and (n − 1)forms are always fiberwise ndimensional and can be identified with vector fields.
Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are
 0 → 1 → 4 → 6 → 4 → 1 → 0;
so the curl of a 1vector field (fiberwise 4dimensional) is a 2vector field, which at each point belongs to 6dimensional vector space, and so one has
which yields a sum of six independent terms, and cannot be identified with a 1vector field. Nor can one meaningfully go from a 1vector field to a 2vector field to a 3vector field (4 → 6 → 4), as taking the differential twice yields zero (d^{2} = 0). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way.
However, one can define a curl of a vector field as a 2vector field in general, as described below.
Curl geometrically
2vectors correspond to the exterior power Λ^{2}V; in the presence of an inner product, in coordinates these are the skewsymmetric matrices, which are geometrically considered as the special orthogonal Lie algebra (V) of infinitesimal rotations. This has (^{n}
_{2}) = 1/2n(n − 1) dimensions, and allows one to interpret the differential of a 1vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) does n = 1/2n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no nontrivial 2vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6dimensional Lie algebra .
The curl of a 3dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2dimensional vector field, as in the examples on this page.
Considering curl as a 2vector field (an antisymmetric 2tensor) has been used to generalize vector calculus and associated physics to higher dimensions.[10]
Inverse
In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W). This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential.
If W is a vector field with curl(W) = V, then adding any gradient vector field grad( f ) to W will result in another vector field W + grad( f ) such that curl(W + grad( f )) = V as well. This can be summarized by saying that the inverse curl of a threedimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law.
See also
Part of a series of articles about 
Calculus 


 Helmholtz decomposition
 Del in cylindrical and spherical coordinates
 Vorticity
References
 Weisstein, Eric W. "Curl". MathWorld.
 Norm ISO/IEC 800002, item 217.16
 Proceedings of the London Mathematical Society, March 9th, 1871
 Collected works of James MacCullagh
 Earliest Known Uses of Some of the Words of Mathematics tripod.com
 Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 9780521861533
 Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 9780071615457
 Gibbs, Josiah Willard; Wilson, Edwin Bidwell (1901), Vector analysis, hdl:2027/mdp.39015000962285
 Arfken, George Brown (2005). Mathematical methods for physicists. Weber, HansJurgen (6th ed.). Boston: Elsevier. ISBN 9780080470696. OCLC 127114279.
 McDavid, A. W.; McMullen, C. D. (20061030). "Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions". arXiv:hepph/0609260.
Further reading
 Korn, Granino Arthur and Theresa M. Korn (January 2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0486411478.
 Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0393969975.
External links
 "Curl", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 "Vector Calculus: Understanding Circulation and Curl – BetterExplained". betterexplained.com. Retrieved 20201109.
 "Divergence and Curl: The Language of Maxwell's Equations, Fluid Flow, and More". June 21, 2018 – via YouTube.