Vortex-free vector field

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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

  1. ^ Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (Eds.): Small encyclopedia of mathematics. Leipzig 1970, p. 549.