I know that not every vector field is a curl of something, and that not every vector field is a gradient of something. However, I want to know if every scalar field (in $\displaystyle R^3$) a divergence of some vector field (not necessarily unique), and is every scalar field a Laplacian of some scalar field?

If not, can a scalar field be decomposed into a sum of two or more scalar fields which are a divergence and/or a Laplacian? (similar to Helmholtz decomposition)

Also, is there a first-order derivative of a scalar field (in $\displaystyle R^3$) that yields a scalar field? I'm not looking for a time derivative or a directional derivative.