Talk:Differential geometry of surfaces/Archive 3

Page on Parametric surface
I just became aware of the page Parametric surface. Should it be absorbed into this one? Gumshoe2 (talk) 15:18, 12 August 2020 (UTC)
 * IMO, this is a question that cannot be resolved without considering the whole structure of our articles about surfaces, which date from April 2016 (I was one of the main contributors of this restructuration). So, we have Surface and Surface (mathematics) that are both broad-concept articles. The latter is aimed to guide readers between the different sorts of mathematical surfaces. Some of the articles that are linked to in Surface (mathematics) by the template main article have not be adapted to the new structure. In particular, the content of Surface (mathematics) is not a summary of Parametric surface.
 * So, aftter some thinking, my opinion is that Parametric surface should be rewritten by merging some sections here and focusing it to elementary properties and generalities that are essentially the same for algebraic surfaces and differentiable surfaces (this is the spirit of Surface (mathematics)).
 * Also, it is worth to think to add to Surface (mathematics) a section on metric properties of surfaces, with Differential geometry of surfaces as a main article. After all this is the main object of this article.
 * Also, several sections of Surface (mathematics) are reduced to a template main article. It would be worth to expand them (They are in this state because I have not the competence to do it myself, or this would need too much work for me). This is the case of the section "Differentiable surface" which should be expanded and possibly renamed. D.Lazard (talk) 16:58, 12 August 2020 (UTC)
 * I see, I hadn't realized there were so many pages. It seems to me like Surface and Surface (mathematics) are both very good pages to have, although of course the latter needs some improvement. Just to have them in one place, it seems like the remaining pages in question are: Surface (topology), Differential geometry of surfaces, Parametric surface, Implicit surface, Riemann surface, Algebraic surface, rational surface. It seems that the last two are mostly about the complex number setting of four-dimensional spaces, so I'll leave them aside as being distinct topics. I believe that the ideal situation would be to have the following core pages:
 * Surface (topology), about two-dimensional topological manifolds and smooth manifolds
 * Riemann surface
 * This page, modified to be purely about differential geometry of surfaces in $ℝ^{3}$, and absorbing the present Implicit surface and Parametric surface, which it nearly does already. It would include the present sections 1-4 together with discussion of the Gauss map, and the present sections 6-8 as adapted to the case of surfaces in $ℝ^{3}$. I would estimate that the optimal length of the core material, aside from some examples, is about 150% the length of the current section 3, which seems to me like a reasonable article length. Perhaps there should also be a brief summary of affine geometry in this setting.
 * I think it's arguable that there should be a full page for two-dimensional Riemannian manifolds. It would presumably include a description of the surfaces of constant curvature, together with the uniformization theorem and the Gauss-Bonnet theorem, all of which are presently on this page. The counterargument, which I would lean towards, would be that each of those topics would make perfect and essentially complete sense on their own pages (space form, uniformization theorem, Gauss–Bonnet theorem). I think the topics could be briefly summarized in a fully satisfactory way in a short section about two dimensions on the page for Riemannian manifold. The other settings in which two-dimensional Riemannian geometry is important, such as in harmonic maps, could be fully dealt with on the individual pages; it doesn't seem to me that a single encyclopedic page would be helpful to anyone.


 * That leaves the pages implicit surface and parametric surface. It isn't clear to me why their scope is limited to two dimensions. I agree that both could be expansions of what they presently are on Surface (mathematics).


 * There's also the page Riemannian connection on a surface. It isn't at all clear to me why it's been made special to two dimensions, since two dimensionality is really only used there to give explanatory pictures, which is already always done in discussing those topics anyway. My opinion is that it could be an excellent page if it were reworked to give an overview of submanifold theory in Riemannian and pseudo-Riemannian geometry, putting the bundle formalism side by side with the tensor calculus. A page for that topic does not seem to exist at the moment, which seems like a major omission. Gumshoe2 (talk) 18:34, 12 August 2020 (UTC)
 * Parametric curves could probably be merged (with care). Mathsci (talk) 08:02, 14 August 2020 (UTC)

Section on orthogonal coordinates
Is there a good reason for its presence? The formula for the Gaussian curvature already appears as part of a list on the page Gaussian curvature. What is the significance of the "classical derivative formula of Gauss"? There should at least be a more precise reference for it, since I couldn't find it in Berger or Eisenhart's books Gumshoe2 (talk) 07:15, 13 August 2020 (UTC)
 * I see now, it's used in the proof of Gauss-Bonnet. A precise reference is page 140 of Spivak's Comprehensive Introduction, vol. 2. On the level of specificity it's given here, it seems like good material, but much more appropriate for the Gauss-Bonnet wiki page. In the end, I believe it should be transferred there. Gumshoe2 (talk) 15:27, 13 August 2020 (UTC)
 * Page 113 of Struik. Page 472 in Appendix C of Michael Taylor's book (connections and curvature): that's the reference in the article. I haven't checked elsewhere. Eisenhart (1909) does discuss orthogonal systems, but it's hard to locate material there. Mathsci (talk) 15:40, 13 August 2020 (UTC)
 * The reference to the "classical derivative formula of Gauss" is to Eisenhart and Berger. I've found it in section 88 of Eisenhart. Gumshoe2 (talk) 15:44, 13 August 2020 (UTC)
 * In the copy of Eisenhart I'm reading (published by Ginn & Co in 1909), section 88 is about the area of a geodesic triangle, page 209-212. Section 64, page 155 gives the required formula for Gaussian curvature (12) with $F$ set to 0. Mathsci (talk) 23:59, 13 August 2020 (UTC)
 * Yes, I was talking about the “classical derivative formula of Gauss”. Anyway, do you have any opposition to moving this material (the Gaussian curvature, the derivative formula, and the application to Gauss Bonnet) to the wiki page for Gauss Bonnet? Gumshoe2 (talk) 00:10, 14 August 2020 (UTC)
 * I do not agree with your proposed move at the moment.


 * Firstly, you're mentioning just one source from 1909. The material you refer to—you write "classical derivative formulas of Gauss"—doesn't exist on Eisenhart (1909, OCR searchable) or Berger (2004). I can see section 88 on areas of geodesic triangles. I can also see Section 64. The treatment of orthogonal nets can also be found in Eisenhart (1940), Section 28. At the moment you have not been able to say where the material comes from. (I've already mentioned Struik and Michael Taylor's Appendix C for the curvature formula when $F$ = 0.)


 * Secondly, he material on geodesic triangles and their area is self-contained and stable. Similarly the material on Geodesic curves on a surface; and Geodesic polar coordinates. All stable. I would also say that computation of geodesic triangles is treated in many textbooks; the presentation has become standard. So the idea of moving the section on orthogonal coordinates there or elsewhere seems unjustified: for potential readers, it's best to keep matters clear. I have not yet been been able to work out what you mean by "the derivative formula". Please could you provide a source? Thanks, Mathsci (talk) 01:55, 14 August 2020 (UTC)
 * ? The phrase "classical derivative formula of Gauss" is from one of only three sentences in the section we're talking about. You inserted the language yourself. The general reference to Eisenhart and Berger's books were likewise added by you. In Eisenhart's book, I have managed to find it in section 88 (specifically page 210, formula (57)). I haven't managed to find it in Berger's book. This material is by you, not by me.


 * It seems that the only reason for including orthogonal coordinates or the "derivative formula" is for the calculation in the section "geodesic triangles". I think it's a good calculation which belongs to a discussion of the proof of the Gauss-Bonnet theorem. It seems extremely natural for that to be on the page Gauss–Bonnet theorem (where it doesn't presently appear), with this page being more appropriate to a summary of results (including, of course, the Gauss-Bonnet theorem for triangles and closed surfaces). It seems to me that that very naturally reflects the difference between a "general summary" page like this one and a "specific theorem" page like that one. Arguing based on how things are presented in books seems very strange, as books are not structured like wikipedia, with distinct pages that can link to one another in this way. Gumshoe2 (talk) 02:42, 14 August 2020 (UTC)

The material or phrase there was original content wholly created by by Arcfrk in January 2008. I simply started to find citations for the uncited content and then incrementally attempted to source the content. There were only six references at that stage: do Carmo (1976), Berger (2004), Singer & Thorpe (1967), O'Neill (1997), Eisenhart (2004), and Han & Hong (2006), all courtesy of Arcfrk. I simply used a harvnb refs to link citations.

For sourcing this section, I checked Appendix C of Michael Taylor. Subsequently, I have checked more carefully that the results are all referenced on pages 112-114 of Struik, sometimes as exercises. do Carmo also cites the first formula on page 240. More specifically, I have verified that Eisenhart covers all the material: in section 63 of Eisenhart (1909), page 153, the differential equations (3) and (4) occur for $H$ and $ω$ = $θ$; and in section 64, page 155, the first equation (12) occurs.

The results on orthogonal nets are a corollary of specializing Gauss' equation in section 64. Those results are just facts which can be recorded, but as an aside or one-off feature. In constast, the standard treatment of surfaces starts with geodesic polar coordinates and then develops everything from there, with the Gauss lemma, the Theorema Egregium, the Gauss-Jacobi equation, the Laplace-Beltrami operator, area of geodesic triangles and then the Gauss-Bonnet theorem. Mathsci (talk) 06:57, 14 August 2020 (UTC)
 * The phrase I was quoting was written by you in the edit I linked above; as I linked above, the references to Eisenhart and Berger were written by you. The mathematical content was written by you on the page that Arcfrk transferred content from to create this page. This entire section seems to be exclusively your work.


 * Your claims about references in Eisenhart and Struik's books are wrong. Equation (3) on page 153 of Eisenhart (and exercise 7 on page 113 of Struik) is about the derivative of the volume form on $S$ and equation (4) (cf. exercise 8 on page 114 of Struik) is about the derivative of the angle between coordinate lines. So although equation (4) is a derivative formula for an angle, it is about a different angle than the one in question here; those equations are about a regular surface in a local parametrization and not about geodesics. The "derivative formula" as given here, of course, is about geodesics. As you say, the equation (12) on page 155 of Eisenhart (and 3-7 on page 113 of Struik) is the Gauss equation, which of course is also not about geodesics.


 * It takes more time for me to correct errors than to create material. Please be more careful with your work. If you do not understand the material fully, please do not insist on writing about it or on keeping your version of it.


 * Please address the second paragraph of my previous message, otherwise I will distribute the material as I see fit. Gumshoe2 (talk) 16:17, 14 August 2020 (UTC)
 * "Your claims about references in Eisenhart and Struik are false." I don't think so. We'll take Struik first.


 * $$K=-{1\over H} \left[\partial_u\left({1\over \sqrt{E}} \partial_u{\sqrt{G}}\right) + \partial_v\left({1\over \sqrt{G}}\partial_v{\sqrt{E}}\right)\right] =

-{1\over 2H} \left[\partial_u\left({1\over H} \partial_u{G}\right) + \partial_v\left({1\over H}\partial_v{E}\right)\right],$$


 * where $$H =(EG)^{1/2}$$. That's the equation in the section of Orthogonal coordinates. You do need to remember the rules for differentiating the square root. Then Exercise 8 on page 114 states, "If ω is the angle between the coordinate curves, show that


 * $$\partial_u\omega= -{D\over E} \Gamma_{11}^2 -{D\over G} \Gamma_{12}^1,\,\,\,\,\partial_v\omega=

-{D\over E}\Gamma_{12}^1 -{D\over G}\Gamma_{22}^1$$."


 * With the assumptions $E$ = 1, $F$ = 0, so that $D$ is the square root of $G$, this is equivalent to the last equation for $$\dot{\omega}$$ (see below).


 * Please assume good faith. I've already noted down the equations for Eisenhart. Equation (12) on page 155 is exactly the formula required with $F$ set to 0. The other two equations in the article follow from equations (3) and (4), assuming that $E$ = 1. We get (with a little care)


 * $$\omega_v = - H_u$$ and $$\omega_u=0$$.


 * Thus we get


 * $${d\omega\over dt} = \omega_u {du\over dt} + \omega_v {dv\over dt} = \omega_v {dv\over dt} = -H_u {dv\over dt}.$$


 * This is the last equation. So no problems with anything from Eisenhart. (You just have to remember that $H$ is the square root of $G$.) The references all work out. The terminology is a little old-fashioned. In this case you appear to have made a mistake. Everybody can make mistakes. Mathsci (talk) 19:03, 14 August 2020 (UTC)
 * In orthogonal coordinates, $ω$, as the angle between coordinate curves, is constantly equal to pi/2. This is the definition of orthogonality. Hence the LHS of the formulas for $∂_{u}ω$ and $∂_{v}ω$ vanish, since the derivative of a constant is zero. It can likewise be directly checked that the RHS are zero. For instance, from the formulas for Christoffel symbols presently in this article, one has in the case $F = 0$ that:
 * $$\begin{align}

\Gamma_{11}^2&=-\frac{1}{2G}\frac{\partial E}{\partial v}\\ \Gamma_{12}^1&=\frac{1}{2E}\frac{\partial E}{\partial v}\\ \Gamma_{12}^2&=\frac{1}{2G}\frac{\partial G}{\partial u}\\ \Gamma_{22}^1&=-\frac{1}{2E}\frac{\partial G}{\partial u} \end{align}$$
 * You can see that, when substituted into the RHS you gave, you indeed get zero. So those formulas, in the present situation of orthogonal coordinates, just say that zero equals zero. I do not believe that this can possibly be equivalent to, or imply, the "classical derivative formula of Gauss." I don't know what went into your statement "with a little care," but it must have been wrong, since $H_{u}$ is not generally zero. If you write out your work I can help you find the error. I do believe you are working in good faith, I just don't believe you fully understand the material. Gumshoe2 (talk) 19:40, 14 August 2020 (UTC)
 * Even just looking indirectly, the argument you've given says nothing about geodesics, which is what the whole formula in question is about. It is (logically) impossible for your argument to be correct. I can't even understand why you'd insist it follows from page 153 in Eisenhart, as I've already told you where to find the actual and exact formula in Eisenhart's book: formula (57) on page 210 in section 88. Gumshoe2 (talk) 20:07, 14 August 2020 (UTC)
 * (ec) The first formuals where $F$ = 0 are fine and can be verified from the source. Arcfrk's original text had:


 * When F=0 in the metric, lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If in addition E=1 and H=G½, then the angle ::::$$\varphi$$  between the geodesic and the line y= constant at their intersection is given by the equation
 * $$\tan \varphi = H\cdot \dot{y}/\dot{x}$$
 * and satisfies the following equation of Gauss:
 * $$ \dot{\varphi} = -H_x \cdot \dot{y}.$$


 * But, checking the second equation, the terms do in fact cancel so that instead of $$\omega_v= -H_v$$, we get $$\omega_v=0$$. Eisenhart formula reads as $$\omega_v= -H\left(E^{-1}\Gamma_{12}^2 + G^{-1} \Gamma_{22}^1\right)$$. But also, from the conditions given by Arcfrk, page 64 of Eisenhart also immediately shows that $$\sin \omega = 1$$ ($F$=0, $E$ = 1). In which case I have no idea where the last equations in the section came from; no idea who dreamed them up. So, wherever it came from, it should be obliterated. The equations with $F$ = 0 are fine. Mathsci (talk) 20:49, 14 August 2020 (UTC)
 * As I posted above, this content was writen by you and Arcfrk just copied it directly to this page. At the moment, you are confusing $ω$, which is defined by a coordinate system, with $φ$, which is defined by a geodesic relative to a polar coordinate system. Anyway, do you now agree with my proposed move of this material to the Gauss-Bonnet theorem page? Gumshoe2 (talk) 20:58, 14 August 2020 (UTC)
 * Also, I keep telling you where the equation comes from. Formula (57) on page 210 in section 88 of Eisenhart. Gumshoe2 (talk) 21:00, 14 August 2020 (UTC)
 * Material you have removed from this article has not always resulted in an improvement. It might not help readers to see what is going on. The rationale that some material generalises to n-dimensions, so could be axed, is too drastic. It's true that Riemannian geometry works in n-dimensions, but surfaces have always been studied first. (NB: your edits to the Cat(k) article left an unexplained image of Birkhoff stranded.)


 * I had completely forgotten what happened with editing to Surface (topology) way back 13 years ago. In early November 2007, I started making edits to the section "differential geometry of surfaces". The split was suggested on 31 January 2008 by Arcfrk, but it was not clean. A few errors crept in in 2007, including the two last sentences about orthogonal systems. In this article, the section on the exponential map describes geodesic normal coordinates. That is probably the best place to mention explicitly geodesic polar coordinates (Wilson, page 145). It is preparation for Gauss' lemma. Mathsci (talk) 04:49, 15 August 2020 (UTC)
 * If your only legitimate grievance is that a prior edit of mine made a picture of Garrett Birkhoff appear out of context, I will proceed with the transfer to Gauss-Bonnet theorem as I see fit. You appear to have misunderstood the argument about derivations and the Levi-Civita connection, which was not simply that the material can be generalized, but rather that it is naturally presented elsewhere, with its presence here not representing any simplification of the material. For reasons which remain unclear, you are rejecting the possibility of adding a sentence with a link to another wiki page, and consequently rejecting the very fundamental difference between a wiki page and a chapter from a textbook. You also seem to still be misunderstanding the situation with the two sentences in the "orthogonal coordinates" section, which are not in error. Gumshoe2 (talk) 10:06, 15 August 2020 (UTC)

As written the section makes little sense to me. Take a Monge form patch, at the origin you have F=0, but at any other point that does not hold, so this does not meet the requirements for an Orthogonal coordinates system. For that you would need some coordinate system based on principle coordinates which do always intersect at right angles (away from umbilics). This is not a problem for the rest of the section which justs uses local calculations.

But I don't see why this an import enough section for a general overview article. Yes, they might be used in a proof of some other result, but that's not sufficient reason for inclusion here. Indeed if we look at Wikipedia talk:WikiProject Mathematics/Proofs there is a debate over when it's appropriate to include a proof at all, and if we were to include a proof then the appropriate place is on the Gauss-Bonnet page not here. --Salix alba (talk): 12:28, 15 August 2020 (UTC)
 * I agree. I see now why your made the edit to the section (which I had reverted) - there is some ambiguity about where $F$ is required to be zero. It is required to be zero throughout the coordinate chart, not just at a single point. I've made a small edit to clarify for the moment.


 * So, just to be clear for the purpose of making a consensus, you agree with my proposed move, which would delete this section entirely from this page as well as the corresponding calculation from the "Geodesic triangles" section? I believe it should certainly be briefly summarized on the page for Gauss-Bonnet theorem, which I think is appropriate since it is not too complex or detailed, which makes it very appropriate for an article on the specific theorem. Gumshoe2 (talk) 12:38, 15 August 2020 (UTC)
 * Yep, sounds a reasonable idea to me. This article needs cutting down.
 * Is there a proof that it is always possible to find a metric where F is everywhere zero? --Salix alba (talk): 12:48, 15 August 2020 (UTC)
 * Yes, the Gauss lemma implies that geodesic polar coordinates are orthogonal. Gumshoe2 (talk) 13:12, 15 August 2020 (UTC)


 * From the results on geodesic polar coordinates and Gauss' lemma, geodesic polar coordinates are orthogonal:
 * $$ ds^2 = dr^2 + G(r,\theta) d\theta^2$$ (Pelham Wilson, page 147). Mathsci (talk) 14:20, 15 August 2020 (UTC)


 * The formula above was already recorded in the article.


 * Actually the equations I wrote down in the article do seem to be fine. They do not depend on using polar coordinates, but could be guessed from them. I have checked Eisenhart's treatment of his 1940 book. He proves a general fact about orthogonal coordinates but with the a hypotheses about geodesics. (32.14) of Eisenhart gives $$d\theta/ds = - \sqrt{g}/g_{11} \Gamma_{12}^2 dv/ds$$ while, by (25.7), $$\tan \theta = dv/du$$. Similarly the same method works for Eisenhart (1909), page 76, (24) and then page 204, (42). Gauss' trick for proving the formula is to use the formula for differentiating arctan combined with the Euler's equations for the geodesic. Mathsci (talk) 19:43, 15 August 2020 (UTC)
 * It is actually explicitly stated in precisely the form you gave in formula (57) on page 210 in section 88. It is true that Eisenhart refers to a polar geodesic coordinate system, but it's clear that in getting from formula (56) to formula (57), he's just using $E = 1$ and $F = 0$ (which are always true in polar geodesic coordinates). This passage from (56) to (57) could be verified by any high-school calculus student. Anyway, I'm glad we finally agree on what's correct and incorrect in the above discussion. Gumshoe2 (talk) 20:41, 15 August 2020 (UTC)
 * As aleady disccussed it's also listed earlier by Eisenhart in the two books. In 2007 I remember taking the metric $$ds^2 = du^2 + G(u,v) dv^2$$ and solving this as an exercise. You can see that from the tentative calculations on Surface (topology). I differentiated arctan and used the Euler equations. Of course I already knew the result for the special case of geodesic polar coordinates, so that provided some guidance. The content was written very rapidly in November 2007 in Aix-en-Provence. I had been in the Schrödinger Institute just before that and was in UC Berkeley after that. The main thing is that Eisenhart added the initials C.G. and 1827 in his treatise lest we forget. BTW as you write the passage from (56) to (57) is a triviality, not even a high school exercise. The passage from $$\tan \theta=H {\dot{y}\over \dot{x}}$$ to the formula for $$\dot{\theta}$$ is a bit trickier. Mathsci (talk) 21:55, 15 August 2020 (UTC)
 * For anyone reading along who would like to verify for themselves, it is quite easy: you just compute the Christoffel symbols in the special case $E = 1$ and $F = 0$, and put them into the geodesic equation to get
 * $$\ddot{x}=\frac{1}{2}\frac{\partial G}{\partial u}\dot{y}^2,\quad\ddot{y}+\frac{1}{G}\frac{\partial G}{\partial u}\dot{x}\dot{y}+\frac{1}{2G}\frac{\partial G}{\partial v}\dot{y}^2=0.$$
 * The angle formula for an inner product space says that the cosine of the angle in question equals
 * $$\frac{\dot{x}}{\sqrt{\dot{x}^2+G\dot{y}^2}}.$$
 * By trigonometry, this is the angle in a right triangle whose opposite leg is $$\sqrt{G}\dot{y}$$ and whose adjacent leg is $$\dot{x}.$$ The problem is now an "instantaneous rate of change" problem like you have in college calculus. The geodesic equations encode the rates of change of the leg lengths, and you have to find the rate of change of the angle. You have to use the chain rule to get the rate of change of $$\sqrt{G(x,y)}.$$ There's nothing remotely "painful" about this calculation. Personally, I can't believe it's taken two days of continuous conversation to get to the point where Mathsci will recognize the phrase "classical derivative formula of Gauss" out of three sentences in the section he wrote, to not recognizing that it is proved in section 88 of Eisenhart, to insisting on a false verification of it based on other material, to then saying that the statement is wrong and should be removed, finally to then doing the very simple calculation by himself and recognizing that it is correct. I apologize for sounding aggrieved and I apologize to all other readers of the talk page for the space this has taken up. I don't know how else to deal with it other than by responding to all of the mistaken points. Gumshoe2 (talk) 02:18, 16 August 2020 (UTC)


 * More running commentary? I used the word "painful" to describe edits made in November 2007, 13 years ago. Back then, I remember a private wiki-meetup with admin Elonka in Aix. There were also wiki-meetups in Cambridge, almost always involving Charles Matthews. He even posted a wiki-image of me. There was also a meetup at The Free Press, Cambridge: the participants, some of whom arrived late, were all mathematicians. The manipulation on pages 173-174 of Eisenhart's 1940 book is publicly available in an OCR readable version. I don't see any "false verification," whatever that's supposed to mean. Please assume good faith. Are you not possibly getting the wrong end of the stick?  Mathsci (talk) 05:29, 16 August 2020 (UTC)
 * Your sentence was literally "There was one painful but ultimately correct calculation: I had to refer to a Princeton treatise by Eisenhart from 1909 and then double check with a 1940 text also by Eisenhart." And, as I said before, I believe you are editing in good faith but do not have a very good understanding of some of the material. I believe this whole section is proof of that. If you'd like to discuss it further, we should do it on a user talk page so as to not further take up space here. Gumshoe2 (talk) 12:49, 16 August 2020 (UTC)


 * I've already told you that Eisenhart's 1940 book and page number are fine and accurate. Esienhart considers orthogonal nets: these are more general than geodesic polar coordinates. Perhaps it might be an idea to stop making subjective comments about my level of understanding; and to stop seeking proof that it is poor. An administator started discussing mathematics matters with me today; he was talking about a new article in the abstruse area of Graph C*-algebras. (He's a theoretical physicist.) I gave examples of articles that I had worked on. You seemingly ignored the administrator's response. Mathsci (talk) 14:16, 16 August 2020 (UTC)
 * I don't think this is an appropriate use of the talk page. Please take it to a user talk page if you want to continue. Gumshoe2 (talk) 14:28, 16 August 2020 (UTC)
 * You wrote above, "And, as I said before, I believe you are editing in good faith but do not have a very good understanding of some of the material. I believe this whole section is proof of that." There is no need for comments like that. Thanks. Mathsci (talk) 14:44, 16 August 2020 (UTC)
 * Just trying to be clear about where I was coming from. I apologize if it was out of place. Gumshoe2 (talk) 14:50, 16 August 2020 (UTC)

Draft:Calculus on Euclidean space
(I mentioned this at the talkpage of the Wikipedia math project but I'm repeating it in case it wasn't noticed.)

I just want to say I started Draft:Calculus on Euclidean space by copying the section on function in 2 variables. I do agree with other editors that some materials added recently by Mathsci are not specific to surfaces and look out of place here. I think that article can be a better place for those materials. -- Taku (talk) 00:28, 17 August 2020 (UTC) If someone needs to respond, please do that at the talkpage of Wikipedia math project. -- Taku (talk) 00:50, 17 August 2020 (UTC)
 * Seems like a good idea, if it that content doesn't already exist on wikipedia. The case of $n$ dimensions should also be covered. Taylor series can be given in general, using the usual notations for $$\partial^\alpha$$ and $$\alpha!$$. It's probably worth spelling out the two-dimensional Taylor expansion in detail. (For functions of 2 variables, there is the 1907 book of E. W. Hobson.) So far the draft you've proposed is quite close to the beginning pages of Lars Hörmander's "Analysis of Linear Partial Differential Operators", Vol.I. Mathsci (talk) 06:44, 17 August 2020 (UTC)

Brief comment on Calabi's "Basics on the Differential Geometry of Surfaces"
These notes of Eugenio Calabi can be found here. They were originally lecture notes by Calabi, written in 1994. They form pages 585-684 of Chapter 20 in "Geometric Methods and Applications" (2011), Springer, edited by Jean Gallier. Mathsci (talk) 14:33, 16 August 2020 (UTC)
 * It seems then that the reference should be to "Geometric Methods and Applications" and Jean Gallier (who wrote the book, not edited it). The preface of the book says "I am very grateful to Professor Calabi for allowing me to write up his lectures on the differential geometry of curves and surfaces given in an undergraduate course in Fall 1994 (as Chapter 20)." I believe this doesn't merit mention in the bibliography here, as it isn't at all helpful for locating the reference. The existing reference "Calabi, Eugenio (1994), Basics of differential geometry of surfaces," at least, made it basically impossible Gumshoe2 (talk) 14:48, 16 August 2020 (UTC)
 * The pdf file, distributed by UPenn, is listed in my post. There is nothing impossible here; it just needs a bit of initiative. Mathsci (talk) 06:56, 17 August 2020 (UTC)
 * I am saying that the reference should be
 * "Gallier, Jean. Geometric methods and applications. For computer science and engineering. Second edition. Texts in Applied Mathematics, 38. Springer, New York, 2011. xxviii+680 pp. ISBN: 978-1-4419-9960-3"


 * and not
 * "Calabi, Eugenio (1994), Basics of differential geometry of surfaces"


 * Hope this clarifies. Either way, it doesn't seem like a very necessary reference to include, since it was only used on the phrase "There are many ways to write the resulting expression, one of them derived in 1852 by Brioschi using a skillful use of determinants:" which already has a reference anyway, and surely could also be referenced with a number of the other existing standard sources Gumshoe2 (talk) 09:37, 17 August 2020 (UTC)
 * Everybody knows how to use harvnb and citation for formatting. Was there any point in commenting? Mathsci (talk) 12:29, 17 August 2020 (UTC)