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Invector calculus, thecurl, also known asrotor, is avector operator that describes theinfinitesimalcirculation of avector field in three-dimensionalEuclidean space. The curl at a point in the field is represented by avector whose length and direction denote themagnitude and axis of the maximum circulation[1] respectively. 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 calledirrotational. The curl is a form ofdifferentiation for vector fields. The corresponding form of thefundamental theorem of calculus isStokes' theorem, which relates thesurface integral of the curl of a vector field to theline integral of the vector field around the boundary curve.
The notationcurlF is more common in English-speaking countries. In the rest of the world, particularly in 20th century scientific literature, the alternative notationrotF is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use thecross product notation with thedel (nabla) operator, as in,[2] which also reveals the relation between curl (rotor),divergence, andgradient operators.
Unlike thegradient anddivergence, curl as formulated in vector calculus does not generalize simply to other dimensions; somegeneralizations 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; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator ofgeometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensionalcross product, and indeed the connection is reflected in the notation for the curl.
The name "curl" was first suggested byJames Clerk Maxwell in 1871[3] but the concept was apparently first used in the construction of an optical field theory byJames MacCullagh in 1839.[4][5]
The curl of a vector fieldF, denoted bycurlF, or, orrotF, is an operator that mapsCk functions inR3 toCk−1 functions inR3, and in particular, it maps continuously differentiable functionsR3 →R3 to continuous functionsR3 →R3. It can be defined in several ways, to be mentioned below:
One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if is any unit vector, the component of the curl ofF along the direction may be defined to be the limiting value of a closedline integral in a plane perpendicular to divided by the area enclosed, as the path of integration is contracted indefinitely around the point.
More specifically, the curl is defined at a pointp as[6][7]where theline integral is calculated along theboundaryC of theareaA containing point p,|A| being the magnitude of the area. This equation defines the component of the curl ofF along the direction. The infinitesimal surfaces bounded byC have as theirnormal.C is oriented via theright-hand rule.
The above formula means that the component of the curl of a vector field along a certain axis is theinfinitesimalarea density of the circulation of the field in a plane perpendicular to that axis. This formula does nota priori define a legitimate vector field, for the individual circulation densities with respect to various axesa priori need not relate to each other in the same way as the components of a vector do; that theydo indeed relate to each other in this precise manner must be proven separately.
To this definition fits naturally theKelvin–Stokes theorem, as a global formula corresponding to the definition. It equates thesurface integral of the curl of a vector field to the above line integral taken around the boundary of the surface.
Another way one can define the curl vector of a functionF at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosingp divided by the volume enclosed, as the shell is contracted indefinitely aroundp.
More specifically, the curl may be defined by the vector formulawhere the surface integral is calculated along the boundaryS of the volumeV,|V| being the magnitude of the volume, and pointing outward from the surfaceS perpendicularly at every point inS.
In this formula, the cross product in the integrand measures the tangential component ofF at each point on the surfaceS, and points along the surface at right angles to thetangential projection ofF. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation ofF aroundS, and whose direction is at right angles to this circulation. The above formula says that thecurl of a vector field at a point is theinfinitesimal volume density of this "circulation vector" around the point.
To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates thevolume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume.
A third way to define the curl emphasizes its interpretation as twice therotation vector of an infinitesimal spherical volume of field. Itcan be shown that the curl fulfillswhere and are points in,,and denotes the ball of radius centered at.
On the other hand, the instantaneous rotation vector of a rigid body with motion field isgiven, for every ball, by the integral mean formula(thus).We see that the curl appears as twice the rotation vector of an infinitesimal spherical volume of field,seen (up to the third order), as a rigid body with motion field equal to.
Whereas the above three definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinearorthogonal coordinates, e.g. inCartesian coordinates,spherical,cylindrical, or evenelliptical orparabolic coordinates:
The equation for each component(curlF)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(x1,x2,x3) are theCartesian coordinates and(u1,u2,u3) are the orthogonal coordinates, then is the length of the coordinate vector corresponding toui. The remaining two components of curl result fromcyclic permutation ofindices: 3,1,2 → 1,2,3 → 2,3,1.
In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curloperator can be applied using some set ofcurvilinear coordinates, for which simpler representations have been derived.
The notation has its origins in the similarities to the 3-dimensionalcross product, and it is useful as amnemonic inCartesian coordinates if is taken as a vectordifferential operatordel. Such notation involvingoperators is common inphysics andalgebra.
Expanded in 3-dimensionalCartesian coordinates (seeDel in cylindrical and spherical coordinates forspherical andcylindrical coordinate representations), is, for composed of (where the subscripts indicate the components of the vector, not partial derivatives):wherei,j, andk are theunit vectors for thex-,y-, andz-axes, respectively. This expands as follows:[8]
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]whereε denotes theLevi-Civita tensor,∇ thecovariant derivative, is thedeterminant of themetric tensor and theEinstein summation convention implies that repeated indices are summed over. Due to the symmetry of theChristoffel symbols participating in the covariant derivative, this expression reduces to thepartial derivative:whereRk are the local basis vectors. Equivalently, using theexterior derivative, the curl can be expressed as:
Here♭ and♯ are themusical isomorphisms, and★ is theHodge star operator. This formula shows how to calculate the curl ofF in any coordinate system, and how to extend the curl to anyoriented three-dimensionalRiemannian manifold. Since this depends on a choice of orientation, curl is achiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.
Suppose the vector field describes thevelocity field of afluid flow (such as a large tank ofliquid orgas) and a small ball is located within the fluid or gas (the center 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 center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.[9]The curl of the vector field at any point is given by the rotation of an infinitesimal area in thexy-plane (forz-axis component of the curl),zx-plane (fory-axis component of the curl) andyz-plane (forx-axis component of the curl vector). This can be seen in the examples below.
Thevector fieldcan be decomposed as
Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linearforce 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 negativez direction. The results of this equation align with what could have been predicted using theright-hand rule using aright-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.
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 linex = 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 right-hand rule, it can be predicted that the resulting curl would be straight in the negativez direction. Inversely, if placed onx = −3, the object would rotate counterclockwise and the right-hand rule would result in a positivez direction.
Calculating the curl:
The curl points in the negativez direction whenx is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the planex = 0.
In generalcurvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fieldsv andF can be shown to be
Interchanging the vector fieldv 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 fieldF (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 coordinatesand this identity defines thevector Laplacian ofF, symbolized as∇2F.
The curl of thegradient ofanyscalar fieldφ is always thezero vector fieldwhich follows from theantisymmetry in the definition of the curl, and thesymmetry of second derivatives.
Thedivergence of the curl of any vector field is equal to zero:
Ifφ is a scalar valued function andF is a vector field, then
The vector calculus operations ofgrad, curl, anddiv are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifyingbivectors (2-vectors) in 3 dimensions with thespecial orthogonal Lie algebra of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and, these all being 3-dimensional spaces.
In 3 dimensions, a differential 0-form is a real-valued function; a differential 1-form is the following expression, where the coefficients are functions:a differential 2-form is the formal sum, again with function coefficients:and a differential 3-form is defined by a single term with one function as coefficient:(Here thea-coefficients are real functions of three variables; thewedge products, e.g., can be interpreted asoriented plane segments,, etc.)
Theexterior derivative of ak-form inR3 is defined as the(k + 1)-form from above—and inRn if, e.g.,then the exterior derivatived leads to
The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives,and antisymmetry,
the twofold application of the exterior derivative yields (the zero-form).
Thus, denoting the space ofk-forms by and the exterior derivative byd one gets a sequence:
Here is the space of sections of theexterior algebravector bundle overRn, whose dimension is thebinomial coefficient; note that for or. Writing only dimensions, one obtains a row ofPascal's triangle:
the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional 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 aRiemannian manifold, or more generallypseudo-Riemannian manifold,k-forms can be identified withk-vector fields (k-forms arek-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on anoriented vector space with anondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism betweenk-vectors and(n −k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchangek-forms,k-vector fields,(n −k)-forms, and(n −k)-vector fields; this is known asHodge duality. Concretely, onR3 this is given by:
Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields:
On the other hand, the fact thatd2 = 0 corresponds to the identitiesfor any scalar fieldf, andfor any vector fieldv.
Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms andn-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and(n − 1)-forms are always fiberwisen-dimensional 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
so the curl of a 1-vector field (fiberwise 4-dimensional) is a2-vector field, which at each point belongs to 6-dimensional vector space, and so one haswhich yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero (d2 = 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 a2-vector field in general, as described below.
2-vectors correspond to the exterior powerΛ2V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as thespecial 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 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we haven =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 2-dimensional 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 non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra.
The curl of a 3-dimensional vector field which only depends on 2 coordinates (sayx andy) is simply a vertical vector field (in thez direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page.
Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions.[10]
In the case where the divergence of a vector fieldV is zero, a vector fieldW exists such thatV = curl(W).[citation needed] This is why themagnetic field, characterized by zero divergence, can be expressed as the curl of amagnetic vector potential.
IfW is a vector field withcurl(W) =V, then adding any gradient vector fieldgrad(f) toW will result in another vector fieldW + grad(f) such thatcurl(W + grad(f)) =V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknownirrotational field with theBiot–Savart law.
