Incompressible, irrotational flow

The basic equations are

$$\nabla \cdot {\bf v} = 0,$$ (5.1)

and

$$\nabla \times {\bf v} = 0.$$ (5.2)

Equation (5.2) is automatically satisfied by setting ${\bf v} = -\nabla f$ and then (5.1) becomes

$$\nabla^{2}f = 0,$$ (5.3)

and the solution to Laplace's equation is required.