Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field ${\displaystyle {\boldsymbol {v}}({\boldsymbol {x}})}$ is:^[1][2]

$${\displaystyle {\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,}$$

where the scalar fields ${\displaystyle \varphi ({\boldsymbol {x}})}$${\displaystyle ,\psi ({\boldsymbol {x}})}$ and ${\displaystyle \chi ({\boldsymbol {x}})}$ are known as Clebsch potentials^[3] or Monge potentials,^[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\displaystyle {\boldsymbol {\nabla }}}$ is the gradient operator.