Prove that if a scalar field $\phi(x,y,z)$ satisfies Laplace's equation:
$\nabla^2 \phi = 0$
$\nabla \cdot \nabla \phi = 0$
then:
$\vec{v} = \nabla \phi$ is irrotational.
irrotational is defined as any vector field where the curl is zero:
$\nabla \times \vec{v} = 0$
Best Answer
This is an identity in vector calculus: "The Curl of a Gradient is always zero"
Thus:
$\vec{v} = \nabla \phi$
taking the curl of both sides, we have:
$\nabla \times \vec{v} = \underbrace{\nabla \times}_{curl} \underbrace{\nabla \phi}_\text{gradient}$
$\nabla \times \vec{v} = 0$
therefore vector field $\vec{v} = \nabla \phi$ is irrotational.