Rotation of a vector field
on a plane
One of the characteristics of a vector field that are invariant under homotopy. Let be a vector field on a domain
of the Euclidean plane
and let
be the angle between
and some fixed direction; the rotation of
will then be the increment of the angle
when going around a closed oriented curve
along which
, divided by
. For instance, if
is a smooth curve of class
, the rotation of the field
(or
) tangent to
(or normal to
) along
is equal to the total curvature of
divided by
; if
is a vector field (with or without isolated singular points) on a domain
, with Jordan boundary
, then the rotation of
on
is equal to the sum of the indices of the singular points of
in the closure of
(cf. Singular point, index of a). The rotation of a vector field remains unchanged during a homotopic deformation of
which does not pass through the singular points of
.
A generalization consists of the concept of the index of a vector field on an
-dimensional manifold
, at an isolated point
of
. It is defined as the degree of
, as a mapping from a small sphere around
to the unit sphere (cf. Degree of a mapping). It is related to the Euler characteristic. See also Poincaré theorem; Kronecker formula.
Comments
Cf. also Rotation number of a curve, which is the rotation of the unit tangent vector field of the curve along that curve.
References
[a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |
[a2] | W. Greub, S. Halperin, R. Vanstone, "Connections, curvature, and cohomology" , 1–3 , Acad. Press (1972) |
[a3] | A. Pollack, "Differential topology" , Prentice-Hall (1974) |
Rotation of a vector field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_of_a_vector_field&oldid=16123