Curl (mathematics)

When rotation is referred to in the vector analysis, a branch of mathematics, a certain differential operator that maps a vector field in three-dimensional Euclidean space with the help of differentiation, a new vector field. Is it, for example, a flow field, so is the rotation for each location twice the angular velocity with which rotates a mitschwimmender body, so how quickly and to what axis it rotates. This relationship is eponymous.

It must, however, not always to a velocity field and act in a rotary motion; For example, the induction law relates the rotation of the electric field.

A vector field whose rotation is everywhere in an area equal to zero, it is called irrotational or, especially in fields of force, conservative. Is the area simply connected, then the vector field is exactly the gradient of a function when the rotation of the vector field in the area under consideration is equal to zero.

The divergence of the rotation of a vector field is zero. Conversely, in simply connected regions a field whose divergence is zero, the rotation of another vector field.

Examples:

• The vector field that indicates the wind direction and speed of a hurricane in any place, has been in the area surrounding the eye a non-zero rotation.
• The vector field that indicates the speed at each point of a rotating disk, at each point has the same non-zero rotation. The rotation is twice the angular velocity,
• The force field that indicates at each point of the gravitational force of the sun on a test particle is irrotational. The force field is the negative gradient of the potential energy of the particle.

Definition of rotation in Cartesian coordinates

Are the Cartesian coordinates of the three-dimensional Euclidean space, and, and the normalized, mutually orthogonal basis vectors, which indicate each point in the direction of increasing coordinates.

The rotation of a three-dimensional, differentiable vector field

Is the three-dimensional vector field

As a rule of thumb may be considered as determinant of a matrix whose first column contains the Cartesian basis vectors, the second the partial derivatives with respect to the Cartesian coordinates, and the third to be differentiated component functions

However, here are the various columns are not vectors of the same vector space.

Let's give the vectors as column vectors of their Cartesian components, then the cross product of the column vector of partial derivatives with respect to the Cartesian coordinates, the nabla operator, with the column vector of the Cartesian component functions

Other coordinate representations of the rotation

Spherical coordinates

If we write the vector field in spherical coordinates as a linear combination

Normalized to unit length vectors

Pointing at any point in the direction of increasing coordinates, then the rotation

Cylindrical coordinates

Are you the vector field in cylindrical coordinates as a linear combination

The vectors

, which show normalized to unit length at every point in the direction of increasing coordinates, then the rotation

Coordinate -free representation of the rotation as the volume derivative

By means of the classic integral set of Stokes, the rotation, similar to the gradient and the divergence ( source density) are presented as volume derivation. This representation has the advantage that it is coordinate- independent. For this reason, the rotation is often defined in the engineering field directly so.

If a region of space with piecewise smooth boundary and volume, then by the rotation of the vector field at the point by means of the volume derivative

Be calculated. This means the outer surface vector element of said outwardly facing surface normal vector and the scalar element. In order to limit the formation region of space is contracted to the point p, so that its contents to zero.

Is replaced by a flow velocity, the rotation appears as a vortex density. Synonyms Similar formed also exist for the divergence ( source density) and the gradient ( force density ). The cross plots of the previous section result from the volume derivative, if one chooses the particular volume element as a region of space.

Axialvektorfeld

The rotation of a vector field is a pseudo- vector field. A vector field is transferred at the reflection in the origin in its negative at the mirrored site, the rotation of the vector field does not change in this mirroring its sign,

Vector field in two dimensions

A vector field in two-dimensional Euclidean space can be used as a vector field

Be considered in three dimensions, which does not depend on the third coordinate, and the third component disappears. Its rotation is not a vector field of this type, but also has a third component,

If one defines the rotation in two dimensions as the differential operator

The result is a scalar function, not a vector field.

Related to the angular velocity

We consider for simplicity the rotation of a rigid body about the axis with constant angular velocity case, the rotation angle increases uniformly with time, and each point traverses a path

The velocity

The velocity field of a rigid rotation about the axis, therefore, is as indicated in the example above,

Its rotation is twice the angular velocity

Decomposition into sources and irrotational part

Dual continuously differentiable vector fields, which sufficiently rapidly go with their derivatives for large distances to zero, it can be clearly divided into an irrotational part and a source-free part,

Thereby indicate the divergence and the gradient, wherein the definition is the usual convention in physics. Mathematically:

This decomposition is part of the Helmholtz theorem.

Calculation rules

The rotation is linear. For all constants and differentiable vector fields and is

The rotation of a vector field vanishes if and only if it is locally a gradient. The divergence of a vector field vanishes if and only if it is locally the rotation of another field,

And the other implications are special cases of the Poincaré lemma.

For differentiable functions and vector fields and the product rules

For the two- fold application of the rotation

For a vector that depends on a scalar, and this from place to 3D applies the chain rule

Integral theorem of Stokes

Main article → Stokes' theorem

The integral over an area about the rotation of a vector field is by the theorem of Stokes equal to the line integral over the boundary curve over

By the double integral is emphasized left that one starts from a two-dimensional integration. On the right side the circle symbol is intended to emphasize the integral sign, that it is an integral over a closed path. The orientation corresponds to the three- finger rule: the following three vectors, namely, first, the vector in the direction of the surface normal, secondly, the vector in the tangential and third, pointing from the edge of the area vector, corresponding to the thumb, index finger and middle finger of the right hand, that is, they form a legal system. Often one writes by highlighting with the direction the size.