What are Vector Fields with Zero Divergence and Curl in 2D?

[Aikon]

Homework Statement

This problem is in Introduction to Eletrodynamics, of Griffiths, 3^rd edition, p.20, problem 1.19. He asks a vector function v(x,y,z), other than the constant, that has:
$\nabla\cdot\vec{v}=0 \mbox{ and } \nabla\times\vec{v}=0$

Homework Equations

I hope you know them.

The Attempt at a Solution

I tried to force the divergence to be zero, using $\vec{v}$, like this: $\vec{v}=v_x(y,z)\hat{x}+v_y(x,z)\hat{y}+v_z(x,y) \hat{z}$
then i solved for the curl of v to be zero and this gave me 3 partial diferencial equations, and so I stopped and decided to get help. Some ideas?

 

[Pengwuino]

One convenient fact that probably will help you is the fact that $\nabla \times (\nabla f(\vec{x})) = 0$.

Let $\vec{v} = \nabla f(\vec{x})$ and your second requirement is automatically satisfied. Then you just need to determine what 'f' is based on the first requirement.

 

[LCKurtz]

Another suggestion -- look for a 2D field F = <u(x,y),v(x,y),0> and think about the Cauchy Riemann equations.