Restriction of a div-free vector field to a plane?

Question

Suppose we restrict a divergence-free vector field on $\mathbb{R}^3$ to some plane. What can be said about the restricted vector field? It no longer has to be divergence-free, of course. But can any vector field on the plane be viewed as the restriction of some div-free field on $\mathbb{R}^3$? And if not, is there a concise way of characterizing the kinds of fields you can get? Thanks.

Answer 1

Can any vector field on the plane be viewed as the restriction of some div-free field on $\Bbb R^3$?

Yep. Suppose we are given a $F = [f_x,f_y] : \Bbb R^2 \to \Bbb R^2$. Define an $\tilde F = [\tilde f_x,\tilde f_y,\tilde f_z]: \Bbb R^3 \to \Bbb R^3$ so that $F(x,y) = [\tilde f_x(x,y,0), \tilde f_y (x,y,0)]$ by setting:
$\tilde f_x(x,y,z) = f_x(x,y)$
$\tilde f_y(x,y,z) = f_y(x,y)$, and $$ f_z(x,y,z) = -z\left( \frac{\partial f_x}{\partial x}(x,y) + \frac{\partial f_y}{\partial y}(x,y) \right) $$ We can quickly verify that $\nabla \cdot \tilde F = 0$.