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 ${\displaystyle {\vec {X}}({\vec {r}})}$

${\displaystyle \oint _{S}{\vec {X}}({\vec {r}})\cdot \mathrm {d} {\vec {s}}=0}$

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 . ${\displaystyle {\vec {X}}({\vec {r}})}$ ${\displaystyle {\vec {X}}({\vec {r}})}$

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 ${\displaystyle {\vec {X}}({\vec {r}})}$

${\displaystyle \mathrm {rot} \ {\vec {X}}({\vec {r}})={\vec {0}}}$, 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 : ${\displaystyle \Phi ({\vec {r}})\ }$

${\displaystyle {\vec {X}}({\vec {r}})=\mathrm {grad} \ \Phi ({\vec {r}})={\vec {\nabla }}\,\Phi ({\vec {r}})}$,

so that also applies:

${\displaystyle \mathrm {rot} \ (\mathrm {grad} \ \Phi ({\vec {r}}))={\vec {0}}}$.

Individual evidence

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