Radial vector field ($\mathbf{F}(\mathbf{r})=f(r)\mathbf{r}$)

A vector field directed along the radius from a fixed origin, with magnitude depending only on radius. In three-dimensional Euclidean space and away from the origin, $\mathbf{F}=f(r)\mathbf{r}=rf(r)\hat{\mathbf{r}}$. The product rule gives $\nabla \cdot \mathbf{F}=3f+r{f}^{\prime }$, and the curl vanishes because the derivative matrix is symmetric. These identities alone do not establish regularity at the origin.