From the Kelvin circulation theorem, one can deduce an importance property. The field where the vorticity is zero \(\nabla \times \mathbf{u} = 0\) (curl \(\mathbf{u} = 0\)) is termed irrotational.
Any vector field \(\mathbf{u}(\mathbf{x})\) whose curl vanishes in a simply-connected domain \(D\) is represented in terms of a potential function \(\Phi(\mathbf{x})\) as
\[
\mathbf{u} = \nabla \Phi = \left( \frac{\partial \Phi}{\partial x}, \frac{\partial \Phi}{\partial y}, \frac{\partial \Phi}{\partial z} \right) = (\partial_x \Phi, \, \partial_y \Phi, \, \partial_z \Phi)
\]
In this sense, an irrotational flow is also called a potential flow. The scalar function \(\Phi(\mathbf{x})\) is called the velocity potential. The flow in which the vorticity is not zero is said to be rotational.
For a potential flow, setting \(\boldsymbol{\omega} = 0\) in the equation of motion \(\frac{\partial \mathbf{u}}{\partial t} + (\nabla \times \mathbf{u}) \times \mathbf{u} = -\nabla \left(\frac{|\mathbf{u}|^2}{2} + h + \chi \right)\), we have
\[
\partial_t \mathbf{u} + \nabla \left( \frac{|\mathbf{u}|^2}{2} + h + \chi \right) = 0
\]
We can integrate this Euler’s equation of motion. Substituting the expression \(\mathbf{u} = \nabla \Phi = \left( \frac{\partial \Phi}{\partial x}, \frac{\partial \Phi}{\partial y}, \frac{\partial \Phi}{\partial z} \right) \)
for \(\mathbf{u}\) and writing \(u = |\mathbf{u}|\)
$$
\nabla \left( \frac{\partial \Phi}{\partial t} + \frac{u^2}{2} + h + \chi \right) = 0
$$
This means that the expression in \((\,)\) is a function of time \(t\) only
$$
\partial_t \Phi + \frac{1}{2} u^2 + h + \chi = f(t)
$$
where \(f(t)\) is an arbitrary function of time