Radial symmetry
Radial symmetry is a form of symmetry in which an object is invariant to all rotations (i.e. all angles and all axes through the center of symmetry). For a reference system , only the origin of the coordinates, but not the orientation, is important if you want to describe a radially symmetrical object. In the three-dimensional case, the radial symmetry is also called spherical symmetry , since spheres (more precisely: also concentric combinations of spherical surfaces) are the only radially symmetrical three-dimensional objects. Physical fields that have radial symmetry, are radial fields mentioned.
Radially symmetric field
Radially symmetric fields play a special role in physics . All radially symmetric fields have in common that they are invariant to linear, length-preserving coordinate transformations . Depending on whether it is scalar fields , vector fields or tensor fields , there are also other properties to uniquely characterize these fields.
Scalar field
A scalar field is radially symmetric if and only if it can be written as a function that only depends on the distance to the coordinate origin:
Vector field
A vector field is radially symmetric if and only if its amounts only depend on the distance to the coordinate origin and the field always points in the radial direction. So a scalar function can be found such that:
It is the corresponding unit vector in the radial direction. An example of a radially symmetrical vector field is the electric field of a point charge .
The gradient of a radially symmetrical scalar field is a radially symmetrical vector field. For example is the gravitational potential
a radially symmetric scalar field. Its gradient, the acceleration of gravity
is the associated vector field.
See also
Remarks
- ↑ a b In both cases o. B. d. A. the origin of the coordinates chosen so that the center of symmetry is located there.