# 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 ${\ displaystyle {\ vec {X}} ({\ vec {r}})}$ ${\ displaystyle \ mathrm {red} \ {\ vec {X}} ({\ vec {r}}) = {\ vec {0}}}$ , 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:

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

## Individual evidence

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