Definition talk:Laplacian/Riemannian Manifold

Two things
a) For all practical purposes we can stick to $\nabla^2$, but we should still mention $\Delta$ somewhere at the bottom, because this abreviation can still be found in various texts.

b) There is no such equivalence. Riemannian case is much more general and includes curved surfaces. The best we can say is that if geometry is flat then there exist coordinates in which $\nabla^2 f = \dfrac {\partial^2 f}{\partial x^2}$. Note that here $\nabla$ denotes covariant derivative (for now we only have def for vector fields, but it can be extended to scalars, tensors etc.), not partial.--Julius (talk) 07:39, 3 October 2022 (UTC)