I am reading Riemannian Geometry and Geometric Analysis by Jost, and this is one of the exercises concering Laplace operators:
3.1.a)
The Laplace operator $\Delta$ on $S^n$ on functions $f: S^n \to \mathbb{R}$ can be obtained from the Euclidean Laplace operator $\Delta_0$ on $\mathbb{R}^{n+1}$ via
$$\Delta f = \Delta_0 f + n \frac{\partial F}{\partial r} + \frac{\partial^2 F}{\partial^2 r}.$$
That is all there is in the exercise, besides a note on polar coodrinates on the sphere $S^n$.
Any idea what $F$ refers to in this equation? I guess it should be some extension of the function $f$ to $\mathbb{R}^{n+1}$, but I guess it could not be $F= f( \frac{x}{\vert \vert x \vert \vert})$, because this would not have a radial derivative.