Curl (mathematics)
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.
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.
In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F 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 (nabla) operator, as in
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
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.
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.
Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p. More specifically, the curl may be defined by the vector formula
To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume.
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.
has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if
is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over.
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 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.
Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.
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 right-hand 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 right-hand rule would result in a positive z direction.
The curl points in the negative z direction when x is positive and vice versa.
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).
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.
The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra
Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div.
On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian 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 k-vectors and (n − k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold.
2-vectors correspond to the exterior power Λ2V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra
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.
Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions.
