Finding a $\mathbb{R}^3$ vector field whose components are with continuous partial derivatives and whose divergence is 1

Question

I’m trying to solve an exercise that asks to find a vector field in $\mathbb{R}^3$ satisfying the following:

• The components of the vector field must have continuous partial derivatives up to at least order 2 on $\mathbb{R}^3$.
• The divergence of the vector field must be 1.

Answer 1

Consider the vector field $F : \mathbb{R}^3 \to \mathbb{R^3}$, defined by $F(x,y,z) = (x,0,0)$. Then $F$ is in fact $\mathcal{C^{\infty}}$, and it is pretty clear that $\text{div}(F) = 1$ identically on all of $\mathbb{R^3}$.