existense of non constant vector valued function f , which is both solenoidal & irrotational

Question

does there exist a non constant vector valued function f , which is both solenoidal & irrotational?

I have tried to solve equations to get a function like this, but I messed it up.Please help. I cant see how to proceed.

Answer 1

I'll asume that you're looking for an example defined on the whole of $\mathbb R^3$. In this case, the vector field $\mathbf F$ is irrotational ($\nabla \times \mathbf F = 0$) if and only if there exists a scalar field $\phi$ such that $\mathbf F = \nabla \phi$.

For $\mathbf F$ to be solenoidal too ($\nabla . \mathbf F = 0$), the condition is that $\phi$ should satisfy Laplace's equation $\nabla^2 \phi = 0$.

There are plenty of solutions to Laplace's equation. A non-trivial example is $\phi = \frac 1 2 (x^2 - y^2)$, which gives $\mathbf F = \nabla \phi = (x, -y, 0)$.