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.

Under way, I'll work on changing pages as appropriate. --prime mover (talk) 08:36, 3 October 2022 (UTC)

Done. Thanks for that --prime mover (talk) 08:48, 3 October 2022 (UTC)

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)

That last bit deserves a page which can be transcluded as appropriate into the various definition pages. That would be brilliant. --prime mover (talk) 08:36, 3 October 2022 (UTC)