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Something seems to be missing here: "... the wave equation describes wave propagation, and the Schrödinger equation in quantum mechanics." — Preceding unsigned comment added by 92.12.18.195 ( talk) 15:54, 19 February 2022 (UTC)
(Disclaimer: I'm neither an experienced Wikipedia editor nor a mathematician). The notation universally used for Laplacian is, it seems to me unfortunate, since it is neither the square of anything nor a function composited with itself. I think a really good aside on this anomalous notation is here: http://physics.ucsd.edu/~emichels/FunkyQuantumConcepts.pdf#page=118 . Would it be appropriate to add a footnote linking to this, or an aside in the article? Ma-Ma-Max Headroom ( talk) 16:57, 28 July 2009 (UTC)
Wikipedia has any standard on this? We see the triangle up a lot (an instance this very page) but the triangle down squared is also there (wave equation, for example). I like the triangle squared, and have a particular dislike for the box representing dalembertian (notably absent in wave equation) — Preceding unsigned comment added by 201.9.205.23 ( talk) 17:00, 13 January 2013 (UTC)
Although heavily used by physicists the notion is misleading, not to say wrong: On a Riemannian manifold with its Levi-Civita connection the Laplace(-Beltrami) operator is defined as the trace of the Hessian. For any smooth function (zero-form) , we first define the (exterior) derivative of as the push-forward (differential) of , i.e. the bundle morphism over , fiberwise defined by the differential . For a vector field and a one-form , the linear connection is acting by
In particular, for the one-form , we call
the Hessian (tensor) of . Finally, the Laplace operator is the contraction of the Hessian, i.e.
More precisely, this means for any orthonormal basis for , we have
Of course, this definition filters though to the case of the special manifold .
So there should be, at least, a hint that the definition is used but not necessarily correct. Wueb ( talk) 20:57, 18 March 2019 (UTC)
It seems to me that Vector Laplacian shouldn't really be its own article. The Laplacian on scalars (as originally defined and as described here) is essentially the same as on vectors, just what we apply it to is different. It feels natural to me that we should have one article to represent the Laplacian as a single concept, which applies equally to scalars and vectors (and tensors, too).
- Ramzuiv ( talk) 03:38, 3 December 2019 (UTC)
Thee section Definition begins as follows:
"The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence () of the gradient (). Thus if is a twice-differentiable real-valued function, then the Laplacian of is the real-valued function defined by:
(1) |
where the latter notations derive from formally writing: "
Regardless of the attempt to justify this at the end, the first equation
(1) |
makes no sense, because there is no operator that can be applied twice to the function to obtain .
This may be fun notation to serve as a mnemonic for the Laplacian operator.
But it does not belong as the first thing that this article says about the definition of the Laplacian.
The section Spectral theory begins as follows:
"The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with: "
Because of how it is written, it sounds as if the spectrum consists only of those among its eigenvalues that satisfy the special condition described by the phrase "for which there is a corresponding eigenfunction ...".
Please correct me if I'm wrong, but isn't this just saying that the spectrum of the Laplace operator consists of the eigenvalues of its negative, ?
(And then mentioning the very definition of "eigenvalues"?)
Furthermore, it should be stated explicitly which topological vector space it is that is understood to be operating on here.