Example of a function $F(x,y)$

Question

I'm trying to find a non trivial function $F(x,y)$ such that $div F(x,y)=0$ everywhere and $F(x,y)=0$ on the unit square.

I know that there are some books that provide such example but I didn't find anything.

Thanks for your help

Answer 1

Take any differentiable function $f(z)$ where $f(z) = 0$ for $|z|\leq 2$ and define $\vec{F}(x,y) = f(x-y)\vec{a}$ where $\vec{a}$ is a constant vector. Then $\vec{\nabla}\cdot \vec{F} = 0$ and $\vec{F}(\text{unit-square)} = 0$ by construction.

An example that should work (and is in fact $C^\infty$) is

$$f(z) = \left\{\begin{array}{cc}0 & |z|\leq 2\\e^{-\frac{1}{(z^2-4)^2}} & |z|\geq 2\end{array}\right.$$

Answer 2

Begin with a $C^2$-function $f:\>{\mathbb R}^2\to{\mathbb R}$ which vanishes identically on the unit square $Q$, but is otherwise completely arbitrary. The vector field $${\bf F}:=*\nabla f:=(-f_y,\>f_x)$$ vanishes identically on $Q$, but is in general not $\equiv{\bf 0}$ on ${\mathbb R}^2$. Furthermore one has $${\rm div}({\bf F})=-f_{yx}+f_{xy}\equiv0\ .$$