Talk:Laplace operator

Rewriting this article

There should really be mention of the Laplace operator on non-Euclidean spaces, e.g. -y^2(f_xx + f_yy) on the upper half-plane.

Yes, the Laplacian should be discussed in a differentiable manifold. Its definition reads $\Delta :=(\mathrm {d} +\mathrm {d} ^{*})^{2}$. This will be a MAJOR rewrite. I'm inclined to undertake this, unless there is resistance.MarSch 16:24, 18 Mar 2005 (UTC)

Be bold! However, it is not clear to me what exactly you want to rewrite. Do you just want to add to this article one or more sections about Δ on manifolds? I think it is good to keep at least the first half of the article the way it is, meaning more elementary, and then get into more complicated issues in sections below, or even in a new article if that new stuff becomes really a lot. Anyway, give it a try, and let us see how it goes. Oleg Alexandrov 17:13, 18 Mar 2005 (UTC)

Ugh. I completely disagree with the form of the recent edits by User:MarSch. As a geometer, I like the fact that the full abstract definition has been added, but it should appear later in the article, after a simpler high-school/college-level definition.

Please keep in mind why people come to Wikipedia in the first place: to learn something new, to refresh thier memory, to look up a forgotten formula. There is nothing worse that one can do to a reader than to overwhelm them with abstractions they don't understand. For example, any chemist, who may have had a few semesters of quantum, would be lost in this article as it currently stands. Ditto for any structural engineer, or electronics engineer. These are people who would use wikipedia, and frankly, they outnumber the geometers by a hundred to one. The article should cater to that level of understanding first, and then, only later, turn to the more abstract definitions. As an example of where this works, see the definition of the discrete laplace operator, which appears at the end of the article, not at the beginining. linas 02:02, 27 Mar 2005 (UTC)

I don't like how the article is turning out either. But, this content would be perfect in an article called Laplacian on manifolds. Oleg Alexandrov 02:48, 27 Mar 2005 (UTC)

I don't like the way it's written now. Certainly it should mention the general versions, but it should begin by saying that in the most concrete cases it's the sum of the second partial derivatives with respect to etc. etc. -- something that everyone (except those who don't know what partial derivatives are) will know what it says. Michael Hardy 03:33, 27 Mar 2005 (UTC)

I agree with you that the article needs to be way better, but I don't think the way to do this is by excluding math, instead we need to add motivation and an informal explanation. I've added some headings which I think are necessary and wrote some flavor sentences, which I hope are close to the truth. Please also see Wikipedia_talk:WikiProject_Mathematics#Structure_of_math_articles MarSch 12:01, 28 Mar 2005 (UTC)

Hi MarSch, no one is trying to exclude math here, nor are we trying to be argumentative jerks. Let me make a simple concrete suggestion: Pull up the version of this article from 6 March, and cut and paste that version above the "formal definition" parts you just added. Thus, the college sophmore will see the traditional ${\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+etc.$ that they expect to see. (This is also what most non-math-major adults expect to see as well, i.e.engineers, etc.). That gives them the warm fuzzies that they are in the right place, reading the right article. Then, as they read further down, they get to your formal definition, and lo, the scales fall from their eyes, and they see the "true" definition. That's all; let the article progress from the simple, concrete, pedestrian definition, to the formal, abstract, exact definition. Hook the reader in with something they can immediately identify with, and then take them on the journey, hoping they'll follow. linas 05:23, 29 Mar 2005 (UTC)

Ok, I have incorporated all relevant material from the 6 march version and I think it is starting to look decent :) MarSch 14:35, 30 Mar 2005 (UTC)

Averages, sign

I think this statement is reversed: "the value of the Laplacian of a function is simply that amount by which a point in space is greater or lesser than the value of the averages of that point's neighbors." it should be something like: "the value of the Laplacian of a function is simply that amount by which the averages of that point's neighbors is greater or lesser than that of the value of a point in space." for instance y=x^2, the Laplacian is 2, while clearly at x=0 the value of the function is less than of the average...

I changed it. Think either way means the same thing, but it's (of course) better not to be confusing. (You can edit articles directly, as well as this discussion page, by the way.) Κσυπ Cyp   17:21, 6 Oct 2004 (UTC)

2? What about 4x^2? lysdexia 13:48, 24 Oct 2004 (UTC)

Which phi

I'm wondering if it's better to use \phi or \varphi:

$\Delta t={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial t \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial t \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}t \over \partial \phi ^{2}}.$

or

$\Delta t={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial t \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial t \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}t \over \partial \varphi ^{2}}.$

or maybe we should write:

$\Delta t(r,\theta ,\varphi )={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}t\left(r,\theta ,\varphi \right)\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial \over \partial \theta }t\left(r,\theta ,\varphi \right)\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2} \over \partial \varphi ^{2}}t\left(r,\theta ,\varphi \right).$

--Jacobolus 05:26, 6 Mar 2005 (UTC)

I would say \varphi looks better. But then you could also consider putting \vartheta instead of \theta. Ultimately it of course does not matter as long as one is consistent. Oleg Alexandrov 05:28, 6 Mar 2005 (UTC)

Actually, some other pages I visited use \phi and \theta. Maybe we should keep things this way for consistency. Oleg Alexandrov 05:32, 6 Mar 2005 (UTC)

Hey. Wow that was a quick response. I think that the norm in mathematics is to use the $\varphi$ symbol instead of $\phi$ I think this looks better. Maybe we should change this across multiple articles?

--Jacobolus 05:45, 6 Mar 2005 (UTC)

You might need to change lots and lots of pages which have to do with spherical and cyllindrical coordinates. I would say you should not rush and ask for more opinions. You might also consider posting this on Wikipedia talk:WikiProject Mathematics where the mathematicians hang around. Oleg Alexandrov 05:51, 6 Mar 2005 (UTC)

Either way, take a look at Nabla in cylindrical and spherical coordinates and whatever leads from there. Oleg Alexandrov 05:51, 6 Mar 2005 (UTC)

Still disagree with recent changes

In spite of Marsch's recent edits, this article still does not look good. Instead of the original nice essay, we now have an incompletely written article, with lots of formulas, and which boldly proclaims that the Laplace operator in Rⁿ is nothing but a particular case of the real thing, which is the Laplace operator on manifolds.

I don't know what Marsch's math biases are, but the Laplace operator in the flat space is very improtant in its own right, and needs an article about it on Wikipedia. This article, in its present form, does not do a good job. What was before, worked well.

I suggest reverting this to the March 6 version. The part about the Laplacian on manifolds, instead of being the center of the article, can become a generalizations section, or even better, a separate article. Oleg Alexandrov 23:58, 1 Apr 2005 (UTC)

The 6 march version is not a "nice essay". It contains little more than the definition and one property. Furthermore it contains redundant data; I don't think it is usefull to include the Laplacian in 2D in the standard basis, when there is already the 3D version and the general flat version in any dimension. It further contains some gibberish about approximating a second derivative. There should be an article about that and it should contain a much better description. I don't see that any information was lost. Also I don't read in the article any bias (bold or otherwise) against the flat version. If you have something to say about the flat version that is interesting/important then you should do it, it will probably have a nice generalization :)

I oppose reverting MarSch 18:28, 2 Apr 2005 (UTC)

You wrote in the article:

The earliest form of the Laplacian is due to Pierre-Simon Laplace. who introduced it when studying <subject>. At the time it was simply a partial differential operator. Thus it worked on flat space. When this was realized, attempts where made to generalize it to arbitrary smooth spaces.

This is why I wrote that for you it matters more what happens on manifolds than what happens in the flat space. Also, the <subject> thing either needs to be filled in, or deleted. Let us make sure the articles have some nice finish. Oleg Alexandrov 18:57, 2 Apr 2005 (UTC)

Now, are you sure that it was Laplace himself who introduced the operator? Oleg Alexandrov 18:59, 2 Apr 2005 (UTC)

No, I am far from sure. This was pure speculation on my part, as explained above, to provide some structure to be filled in and not merely some usefull headings. I was hoping someone would do something with them, so please change them as you see fit. BTW Are you sure this wasn't an April's Fools joke? -MarSch 14:18, 4 Apr 2005 (UTC)

I will do some changes today. By the way, it is good to indeed stick to things we really know about. It is good to hope that somebody else will come and correct that, but from my experience here, that seldom happens, and then misinformation can persist. I will also put some of that numerical stuff back about discretizing the Laplacian. It seems you are a theoretical guy. I am more applied, and the discretization of the Laplacian is a good thing to have. Oleg Alexandrov 15:34, 4 Apr 2005 (UTC)

The proof thing

Hi MarSch. If you want to keep the proof, and in general, the differential geometry part, I think you would need to do some of the following:

(a) explain what d is

(b) explain what d* is

(c) Explain your summation notation (I think you use the Einstein convention, but that is not said)

(d) Explain the * operator which you use as some kind of multiplication.

(e) explain what epsilon is or give a pointer

(f) Explain what $\partial ^{i}$ is as opposed to $\partial _{i}$

(g) Explain what vol is in this context

The way the proof stands now, is several long lines of formulas, and with underfined notation. Also, if not too hard for you, could you please drop here a couple of lines explaining what the usefullness of that proof is, in this context. Thanks. Oleg Alexandrov 00:20, 13 Apr 2005 (UTC)

Hi Marsch, I agree with Oleg here. The proofs are useful, but they should go into a distinct article possibly titled proof of linearity of laplacian or something like that. And yes, you must explain the notation, link to things like Hodge star since otherwise a lot of readers will assume its "some kind of multiplication". (User:Linas forgot to sign).

Well, the proof could even stay in this article, if it is properly explained. Oleg Alexandrov 15:16, 13 Apr 2005 (UTC)

My concern is that the article will become too long. Is the proof going to be something of general value, or is it "merely" establishing the correctness of a formula? Does it illustrate an important concept, or is it a mechanical manipulation of little pedagogical interest? linas 15:36, 13 Apr 2005 (UTC)

This notation should be explained in an article about exterior calculus somewhere, in one place. Not scattered or repeated on each page using it. That should cover (a) to (g). THe proof is just a few lines that don't hurt anybody and that some might find very interesting. I learned a lot doing this stuff. Removing the proof does not eliminate the notation, since it is used in the def and also the derivation of laplacian in general coordinates. And yes it is also a valuable reference. Don't downplay the importance of that. What is the value of any formula that just falls out of the sky and can be edited be anyone? Also proofs should be directly besides what they prove to prevent drifting apart. MarSch 13:27, 14 Apr 2005 (UTC)

You have valid points. As long as you find a way (by referring to some other pages if you wish, or any other way) to explain all the notation, it should be fine. But, I do not agree leaving the proof here with the notation not explained in any way. Oleg Alexandrov 16:23, 14 Apr 2005 (UTC)

And you are right, you used this notation in other places too in this article besides the proof. Even more reasons to have the notation disambiguated. Oleg Alexandrov 17:47, 14 Apr 2005 (UTC)

I've added pointers for most of your points except (c) and (f). It's much better already. (f) should be in metric tensor. (c) yes summation convention and increasing multi-index. Kinda tricky to explain/state. -MarSch 15:29, 18 Apr 2005 (UTC)

I'm having problems with

and d* is the codifferential defined by d* = *d*,

simply because its hard to see that in one case, the star is a superscript, and in the other its not. This just looks hopelessly confusing to the casual reader. How about an alternate notation, such as ${\overline {d}}$ or d⁺ or something like that? Maybe a five-pointed star?

It seems clear enough to me, however I am willing to change it. One solution would be to enforce latex rendering to make the distinction clearer. For an alternate notation we could use δ which is also in use. Will check cochain complex. MarSch 11:45, 19 Apr 2005 (UTC)

Here's another example of the problem with the superscript-star notation: The definition you give here seems to be in notational collision with that in De Rham cohomology. I think you really want to get rid of the superscript-star notation and write $\Delta =d^{+}d+dd^{+}=d*d*+*d*d$ just to make it extra clear and unambiguous. Note that this is also the same as the notation used in the article Hodge star. Please do *not* change the notation used in those two articles; they're fine.

I do not see what is the collision. They use only *d* in the same way as do I. MarSch 11:45, 19 Apr 2005 (UTC)

And, while we're on the topic: I think its is more correct to point out that this operator really should be called the Laplace-Beltrami operator, and that some people call it the Hodge-Laplace operator, and that the word "Laplacian" is usually restricted to that thing defined on Euclidean space.

My book didn't mention this, But they seem to use this terminology in Hodge star. I will create some redirects for those names. Please feel free to include this info. MarSch 11:45, 19 Apr 2005 (UTC)

For the summation convention, just link to einstein summation convention in the article. Explain that's for the italic i. For the summation convention on the capital roman J, make sure you link the ε to Levi-Civita symbol and call it the totally anti-symmetric symbol, and that may be enough to explain the summation.

For point (f), just take the time out and just plain say $\partial ^{i}=g^{ij}\partial _{j}$. Its OK to be wordy here, we're not trying to cram this on a postage stamp for some exam. linas 22:29, 18 Apr 2005 (UTC)

Adding factoids, theorems of general utility

Marsch, could you be persuaded to resurect some of the text from the version of 6 march 2005, dealing with the limit/discrete case? I think it was useful to note that the second derivative (in eucliden space) was f(x+h)-2f(x)+f(x-h) since that leads directly to the topic of digital image filtering with laplacian fileters. Then right after this, it should state that when &Delta f=0 then the average of f over a sphere equals the value in the center, which is I think eye-opening about why the Laplacian is interesting in the first place. (Is this called "Poisson's theorem', maybe??) It might also be nice to add blurbs mentioning heat equation, diffusion equation, schreodinger equation, and poisson equation, since these are the main equations that 90% of readership will be encountering. Also handy to mention that solutions to poissons equation are harmonic functions. linas 22:55, 18 Apr 2005 (UTC)

I agree with Linas that the discrete case should be brought back. And it should be before the differential geometry part, as it still refers to the flat laplacian. The discrete Laplacian is very important in Numerical Analysis.

Linas, can I challenge you to work on this article (both explaining the differential geometry notation, and the discrete thing, and what not)? I think MarSch will not mind us adding new stuff. We can then all work together to integrate things and make this a good article. Oleg Alexandrov 23:12, 18 Apr 2005 (UTC)

Ah, well, yes; but I thought I'd give Marsch first crack. I know that if I spend two hours editing and then Marsch re-writes half of it, I'll just end up getting frustrated. But if the muse comes... I was thinking of working on some special functions today, which came up in an interesting problem recently.linas 23:28, 18 Apr 2005 (UTC)

The stuff you mention sounds interesting. But this is not an article about numerical approximation methods, so I think you should point to one such for your f(x+h)-2f(x)+f(x-h) formula. Then if you could relate this to digital image filtering etc.that would be great. Some of the equations you mention are already in the intro, but the others should definitely also be there. It would probably be best if you first spend two hours adding this stuff, after which I can rewrite half ;) But I certainly don't expect to have to do that. -MarSch 12:00, 19 Apr 2005 (UTC)

Well, a blurb about the numerical laplacian will certainly not hurt. After all, this is not an article about differential geometry either. :) Oleg Alexandrov 15:19, 19 Apr 2005 (UTC)

OK, I'll give it a shot real soon now. But please do not make threats about re-writing half of it before you've seen the result. linas 05:51, 25 Apr 2005 (UTC)

Sorry. That was supposed to be a joke. I thought it was clear, but once again: I do not intend to rewrite anything just yet. MarSch 13:22, 27 Apr 2005 (UTC)

Good luck Linas. I think you are doing MarSch a great favor by tackling the unfinished business of explaining the notation. By the way, if you get a nice long article, one idea would be to fork it into a Laplace-Beltrami operator article. But that's all up to you. Oleg Alexandrov 15:14, 25 Apr 2005 (UTC)

PS If MarSch does not agree with something, I am sure he will first ask here rather than start rewriting. :) Oleg Alexandrov 15:15, 25 Apr 2005 (UTC)

Wow, linas. I think you have done a great job with this article. -MarSch 13:48, 27 Apr 2005 (UTC)

Remark about Dalambertian

Hi Linas. Good job. Just a remark. I would think that the

$\square ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}-{\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}$

which I think is called Dalambertian is not the Laplacian. Certainly not in R^4 with the usual Euclidean structure. I think this goes a bit off topic, what do you think? Oleg Alexandrov 18:18, 26 Apr 2005 (UTC)

Thanks, I'd forgotten that it has a name. The D'Alambertian is the Laplacian for flat Minkowski spacetime; don't think its off-topic, it gives a flavour for what non-euclidean geometry is like. It is recovered directly from the formula for the Laplace-Beltrami by plugging in the flat minkowski metric.

I still plan on adding something very breif on discreteness; I admit the article is now quite long; don't want to split it up, am afraid it will loose continuity. linas 14:32, 27 Apr 2005 (UTC)

That's what I said above, that the D'Alambertian is not the Laplacian in the Euclidean space. So, I am not sure it belongs in the definition part, where one talks about the Euclidean Laplacian. It more likely belongs in the section which actually talks about Laplace-Beltrami, which is below. Oleg Alexandrov 16:55, 27 Apr 2005 (UTC)

But I will leave this up to you. My idea for this article was Part 1: Euclidean Laplacian, and Part 2: Differential geometry Laplacian. But since you are doing all the work anyway, you can decide yourself which way to go. :) Oleg Alexandrov 17:03, 27 Apr 2005 (UTC)

I'd prefer to leave in in the initial section, as a "teaser" for what is to come. Again, college sophmore engineering majors will see the 3D euclidean stuff in class, and have a 50/50 chance of seeing the dalambertian as well, or at least the 4d wave equation (electromagnetism, EE, integrated circuit design etc.). However, these same students will be lost almost immediately in the next section, as it promptly whips into differential forms, and I assume they aren't teaching differential forms to these students. So I'm looking at the dalembertian as a carrot "you too can understand the deeper meaning of the dalembertian, if you apply yourself a bit and wade through the next section" its a sample of what lies ahead, presented in simple terms. linas 22:54, 27 Apr 2005 (UTC)

d and -div adjoints

$\int _{M}df(X)=\int _{M}X(f)=\int _{M}{\mathcal {L}}_{X}(f)=\int _{M}(-1)^{?}*{\mathcal {L}}_{X}(*f)=$

$\int _{M}(-1)^{?}*(f{\mathcal {L}}_{X}\omega +\omega X(f))=\int _{M}(-1)^{?}*(f\mathrm {div} (X)\omega +\omega X(f))=$ $\int _{M}(-1)^{?}(f\mathrm {div} (X)+X(f))=?-\int _{M}f\mathrm {div} X\;$

I thought this is all legal. If ? is odd it would mean we are missing a factor 2: strange. Otherwise it would mean that the last term is zero and break the claimed adjointness: very strange. What am I missing? MarSch 14:23, 27 Apr 2005 (UTC)

Hmm, not to be glib, but maybe a factor of ω ? I'm late to work, I'm rushing, but I'll think about this a bit. I'm thinking that the equation should have been

$\int _{M}(\omega f\mathrm {div} (X)+\omega X(f))=0$

and I wrote the integral without the "implicit" ω. linas 14:46, 27 Apr 2005 (UTC)

Yes, without that the formula makes no sense. My new calculation shows you are right for manifolds without boundary.

$\int _{M}(f\mathrm {div} (X)+X(f))\omega =\int _{M}(f{\mathcal {L}}_{X}+{\mathcal {L}}_{X}(f))\omega =\int _{M}{\mathcal {L}}_{X}f\omega =\int _{M}\mathrm {d} \iota _{X}f\omega =\int _{\partial M}\iota _{X}f\omega$

I think it is better to include this short calculation. -MarSch 15:27, 27 Apr 2005 (UTC)

I'd prefer to leave the short calculation out; I think it makes the article harder to read without conveying much info. The reader has to burn brain cells to undertand the formula, and then when gets to the end, and probably realizes "oh I don't care, anyway". What we really need to do is to establish some sort of wikipedia house style/house convention for supplying proofs or at least details for formulas like this, without cluttering the main article. That way, the casual reader is not impeded by dense formulas, whereas someone who is worried about detail and correctness can check explore more deeply. I will create an experimental page to do this shortly; lets see "how it feels", and if such a format could work out.

I agree in leaving the calculation out. This elementary article is gradually becoming a differential geometry monster. :) Oleg Alexandrov 23:34, 27 Apr 2005 (UTC)

BTW, the surface does not have to be boundary-less; the only requirement is for the support for f to be compact: if support is compact, then you can find an open cover on whose boundary f vanishes, and so the integral on boundry is zero. I really want to keep this adjoint-ness property really really simple, I really just want to say "div is just the transpose of d" and make the point that these two things are really almost exactly the same thing, just kind-of reflected.linas 23:26, 27 Apr 2005 (UTC)

BTW, I have to mention this, just in case: I suspect you might be bothered by the idea of making a true statement that holds for all X and f with compact support, and then wildly jumping to the conclusion that this somehow holds for the operators div and d themselves. If this troubles you, you'd be justified, there's subtle stuff going on, and Sobolev spaces are part of what you'd want to resolve this. But that is way deeper than we want to go here. I just want to say d and div are 'the same thing, sort-of' without diving into something else. linas 23:26, 27 Apr 2005 (UTC)

Doesn't this duality hold more generally for d and δ? -MarSch 15:23, 5 May 2005 (UTC)

partials

I notice that although there is still the def

$\partial _{i}={\frac {\partial }{\partial x^{i}}}$

it is not used immediately following. Not even in the Laplace-Beltrami section. I would prefer to at least use it there exclusively. If you want to keep it before that then we should move this def down. -MarSch 15:41, 27 Apr 2005 (UTC)

Hmm, I'd rather stick to partial/partial x notation, just to keep things concrete, and not force the reader to make irrelevent abstractions. The reader who already knows that partial_i = partial/parital x^i is not going to be impeded by the more verbose notation. On the other hand, there will be readers who will suspect that maybe partial_i is maybe the same thing as partial/parital x^i, but they're not sure. Even when we explain that "hey duude partial_i = partial/parital x^i" they'll still be thinking there's some conspiracy, some depth or theorem there that they don't understand. To ask them to juggle that uncertainty at the same time they're juggling the other uncertainties is a bit too much for an encyclopedia article. I know that math books and teachers love to force thier students to juggle as much as possible ... that's how you get good. But I think that's bad style for an encyclopedia or reference work. linas 23:37, 27 Apr 2005 (UTC)

I would much prefer the simpler notation of $\partial _{i}$ especially when the formulas start to get more complicated. It is easier to see whether the index is up or down. Also this is standard notation. I especially dislike the use of ${\frac {\partial }{\partial x_{i}}}$. I don't think I've seen this before. I really think we should use the simpler notation starting from heading "Laplace-Beltrami operator". -MarSch 13:08, 2 May 2005 (UTC)

I saw the ${\frac {\partial }{\partial x_{i}}}$ used a lot though. I think all of us have different biases depending on our background. But I don't care either way. My bigger problem, MarSch, is that you did not yet explain the $\partial ^{i}$ notation. Hope you find time to do it sometime. :) Oleg Alexandrov 14:56, 2 May 2005 (UTC)

Agree with Linas's changes

The article looks much cleaner now without the proofs (the proofs were not very well written anyway). Oleg Alexandrov 17:34, 18 May 2005 (UTC)

well, I don't like it. But since this is not particular to this article will discuss somewhere else. -MarSch 11:12, 19 May 2005 (UTC)

Elliptic?

Is the Laplacian an elliptic operator or not? In the introduction, it says that it is, but then in the article it is stated that it can be defined on non-Euclidean spaces, like the d'Alembertian. --Joke137 23:07, 18 May 2005 (UTC)

Well, see the discussion above, at Talk:Laplace operator#Remark about Dalambertian. You have a good point. Let us see what Linas has to say. Oleg Alexandrov 23:29, 18 May 2005 (UTC)

Ahh, that'll teach me not to read the whole talk page. It's a problem of semantics more than anything, but one I worry about. Is the operator $\nabla ^{2}$ on a curved space with a pseudo-Riemannian (i.e. $(-,+,\dots ,+)$) metric properly called the Laplacian, d'Alembertian, both, or neither? --Joke137 01:03, 19 May 2005 (UTC)