Velocity potential

Remarks

A field with a velocity potential is a conservative vector field. If $\mathbf{u}$ is continuously differentiable and has a potential, then its vorticity $\nabla \times \mathbf{u}$ is zero. Conversely, on a simply connected domain every continuously differentiable irrotational field has a potential, unique up to an additive constant. If moreover the flow is incompressible, then $\Delta \phi = \nabla \cdot \mathbf{u} = 0$, so $\phi$ solves Laplace's equation. Example: $\phi = (x^2 - y^2)/2$ is a potential for $\mathbf{u} = (x, -y)$; check: $\phi_x = x$ and $\phi_y = -y$. Non-example: the rigid rotation $\mathbf{u} = (-y, x)$ has vorticity $2 \ne 0$, so it has no potential. The point vortex $\mathbf{u} = (-y, x)/(x^2 + y^2)$ is irrotational on $\R^2 \setminus \set{0}$ but has no single-valued potential there, since its circulation around the origin is $2\pi$; the domain is not simply connected.