Vortex-free vector field
In physics and potential theory, a vector field in which the curve integral is referred to as vortex free or conservative
for any self-contained boundary curve S always returns the value zero. Is indicated as the force field , the ring is integral along the whole of the boundary curve S against the force was doing work .
Vortex-free are z. B. the static electric field and the gravitational field , but also fields like the velocity field of a potential flow .
If there is no vortex, then applies
- , d. H. the rotation of the vector field is zero (naming).
If the domain is simply connected , the converse also applies.
Vortex-free vector fields can always be formulated as the gradient of an underlying scalar field :
- ,
so that also applies:
- .
Individual evidence
- ^ Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (Eds.): Small encyclopedia of mathematics. Leipzig 1970, p. 549.