Concentric hyperspheres in the n-dimensional Euclidean space $mathbb{R}^n$ are the level hypersurfaces of a radial function f : $mathbb{R}^n{ ightarrow}mathbb{R}$. The magnitude $||{ abla}f||$ of the gradient of such a radial function f : $mathbb{R}^n{ ightarrow}mathbb{R}$ is a function of the function f. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function f : $mathbb{R}^n{ ightarrow}mathbb{R}$ with isolated critical points is a function of f itself, then f is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $mathbb{R}^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.