Curl (mathematics)
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional 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 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 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 notation is more common in English-speaking countries. In the rest of the world, particularly in 20th century scientific literature, the alternative notation is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del operator, as in which also reveals the relation between curl, divergence, and gradient operators.
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; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional 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 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839.
Definition
The curl of a vector field, denoted by, or, or, is an operator that maps functions in to functions in, and in particular, it maps continuously differentiable functions to continuous functions. 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 of along the direction may be defined to be the limiting value of a closed line 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 point as
where the line integral is calculated along the boundary of the area containing point p, being the magnitude of the area. This equation defines the component of the curl of along the direction. The infinitesimal surfaces bounded by have as their normal. is oriented via the right-hand rule.
The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately.
To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface 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 function at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing divided by the volume enclosed, as the shell is contracted indefinitely around.
More specifically, the curl may be defined by the vector formula
where the surface integral is calculated along the boundary of the volume, being the magnitude of the volume, and pointing outward from the surface perpendicularly at every point in.
In this formula, the cross product in the integrand measures the tangential component of at each point on the surface, and points along the surface at right angles to the tangential projection of. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of around, and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point.
To this definition fits naturally another global formula which equates the volume 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 the rotation vector of an infinitesimal spherical volume of field.
It can be shown that the curl fulfills
where 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 is given, for every ball, by the integral mean formula
.
We see that the curl appears as twice the rotation vector of an infinitesimal spherical volume of field,
seen, 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 curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates:
The equation for each component can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1.
If are the Cartesian coordinates and are the orthogonal coordinates, then
is the length of the coordinate vector corresponding to. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.
Usage
In practice, the two coordinate-free definitions described above are 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 has its origins in the similarities to the 3-dimensional 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 3-dimensional Cartesian coordinates, is, for composed of :
where,, and are the unit vectors for the -, -, and -axes, respectively. This expands as follows:
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
where denotes the Levi-Civita tensor, the covariant derivative, is the determinant of the metric tensor 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 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 in any coordinate system, and how to extend the curl to any oriented three-dimensional 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
Suppose the vector field describes the velocity field of a fluid flow and a small ball is located within the fluid or gas. If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis 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.The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane, zx-plane and yz-plane. This can be seen in the examples below.
Example 2
The vector fieldcan 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 direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.
Example 3
For the vector fieldthe curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line, 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 negative direction. Inversely, if placed on, the object would rotate counterclockwise and the right-hand rule would result in a positive direction.
Calculating the curl:
The curl points in the negative direction when is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane.