Clebsch representation

In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Clebsch representation of an arbitrary <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> <a href="/facts/Vector_field/ccFNuPPp">vector field</a> 
 
 
 
 
 v
 
 (
 
 x
 
 )
 
 
 {\displaystyle {\boldsymbol {v}}({\boldsymbol {x}})}
 
 is:

 
 
 
 
 v
 
 =
 
 ∇
 
 φ
 +
 ψ
 
 
 ∇
 
 χ
 ,
 
 
 {\displaystyle {\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,}

where the <a href="/facts/Scalar_field/8MtPTGob">scalar fields</a> 
 
 
 
 φ
 (
 
 x
 
 )
 
 
 {\displaystyle \varphi ({\boldsymbol {x}})}

,
 ψ
 (
 
 x
 
 )
 
 
 {\displaystyle ,\psi ({\boldsymbol {x}})}
 
 and 
 
 
 
 χ
 (
 
 x
 
 )
 
 
 {\displaystyle \chi ({\boldsymbol {x}})}
 
 are known as Clebsch potentials or Monge potentials, named after <a href="/facts/Alfred_Clebsch/NV4IyEAk">Alfred Clebsch</a> (1833–1872) and <a href="/facts/Gaspard_Monge/JMvYFCFv">Gaspard Monge</a> (1746–1818), and 
 
 
 
 
 ∇
 
 
 
 {\displaystyle {\boldsymbol {\nabla }}}
 
 is the <a href="/facts/Gradient/TsR7uukj">gradient</a> operator.

Clebsch representation open-in-new

Clebsch representation