Talk:Calculus on Euclidean space

The intended content

Would it be correct to say that this article is meant to cover chapters 9 and 10 of Rudin "Principles of Mathematical Analysis" and Spivak "Calculus on Manifolds"? If so, the intended topics would seem to be:

• implicit and inverse function theorem
• Taylor's theorem in multiple dimensions
• integration in multiple dimensions
• smooth embedded submanifolds of Euclidean space
• integration of differential forms and the change of variable formula for integration
• Stokes theorem and its relatives

Is this missing anything? It seems that there are already pages for each of these topics - all of which could be improved with better expositions and added content. So what exactly is the distinct purpose of this page? Even in terms of summaries, there is already a page for multivariable calculus, which seems to me to be synonymous with calculus on Euclidean space Gumshoe2 (talk) 09:26, 17 August 2020 (UTC)

• From Talk:Differential geometry of surfaces. "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. -- TakuyaMurata"

• I replied "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."

I'll add now that I think Fubini's theorem in several variables is another topic. It's used anyway in one of the proofs of symmetry of second derivatives (and versions of the Stone-Weierstrass theorem). Proofs of inverse function theorem are also on wikipedia (I inserted one); the implicit function theorem is a corollary. Most of the proofs have been taken from Hörmander.

Calculus is a fairly elementary topic. The other topics are of a different flavour, topics from differential topology (Guillemin & Pollack), way beyond elementary calculus. It's quite unrealistic to write a low level article that combines both elementary calculus and differential topology.

As I've written to Taku, his suggestion on calculus was a very idea and the skeleton he suggested is excellent. Mathsci (talk) 13:19, 17 August 2020 (UTC)

It seems that the current content in the draft could very easily and naturally be included in other articles.
There is, at present, no differential topology.
There is also an existing page for Fubini's theorem which could be used (and is probably where some of the material in Symmetry of second derivatives should be put - that page is almost certainly the wrong place to go into the proof of Fubini). Gumshoe2 (talk) 15:39, 17 August 2020 (UTC)

To respond to @Gumshoe2:, I think the key difference is the use of more abstract point view. For example, in multivariable calculus, a tangent vector is introduced in a manner similar to that in one-variable calculus (i.e., as a derivative). But, in differential geometry, a tangent vector is defined in more intrinsic manner; i.e., a derivation on the germ of functions. Such an abstract point of view will be distracting in the "multivariable calculus" article. Also, I don't think a differential form is typically a part of multivariable calculus; at least, textbooks on multivariable calculus I have used to teach (Stewart is one used in my school) do not cover differential forms; in particular, Stokes in terms of differential forms. I suppose it's a matter of the use of the term "multivariable calculus"; for me, "multivariable calculus" has an image of a typical undergraduate calculus in the universities in the united states. For me, those chapters in Rudin's text are *not* part of multivariable calculus. I do agree there will be a lot of overlaps; I don't think that's necessarily bad. It is usually more convenient for the readers if they didn't have to read several separate articles. As I see, Wikipedia can have two types of math articles; a theory-type one and a topic-specific one. Both types of the articles are needed and are useful for the readers. -- Taku (talk) 04:14, 18 August 2020 (UTC)

In my opinion it’d be much easier to justify an article on calculus on manifolds in Euclidean space, as in Spivak’s book. Anyway, it might be helpful to be specific. What’s to be gained from discussing Taylor’s theorem here as opposed to on its own page? Gumshoe2 (talk) 04:26, 18 August 2020 (UTC)

I think calculus on manifolds can/should redirect to this article; it seems natural to explain how ordinary calculus (of functions and forms) generalizes to calculus (of functions and forms) on manifolds. The benefit of having an article like this is that the readers need not read separate articles on individual articles and also treatment here will be more concise. It’s like this: it is standard to have an article on modern European history in addition to articles on French history, German history etc. Sure there are overlaps; that’s not necessary bad. Taylor’s theorem will have very detailed discussion (as it should), while the article like this one can focus on connections and relations between different topics; for example, a differential form is (by definition) a skew-multi-linear function on tangent vectors and so it will be convenient for the readers to see both definitions of tangent vectors and differential forms. —- Taku (talk) 22:43, 18 August 2020 (UTC)

Sure, I understand the idea. But what would you say about Taylor’s theorem here that you couldn’t easily say on its own page? French history is quite complex; I think Taylor’s theorem isn’t! Gumshoe2 (talk) 23:03, 18 August 2020 (UTC)

Taylor's theorem can be fairly nontrivial (follows the link) especially regarding on differentiability and remainder terms. The readers indeed prefer to read a single article instead of being asked to keep following links. In fact, many complains on math articles are that to read an article, you first have to follow the links to know the definitions, notations, statements of theorems etc. It is thus *encouraged* to have some degree of duplications. So I am not sure what is wrong with having a mention of Taylor's theorem here. To be clear, I am not planning to expand the discussion of Taylor expansion we already have here (initially written by Mathsci). Rather I think the current section on functions of two variables is overly detailed and can be simplified. -- Taku (talk) 00:01, 19 August 2020 (UTC)

I am also open to omitting the discussion of Taylor's theorem altogether. While I don't think it hurts, I also don't see that's an integral part of the article like this. -- Taku (talk) 00:14, 19 August 2020 (UTC)

As done in Hörmander's book (LPDO I, page 13), there is a precise formula for Taylor's theorem This generalises to n dimensions using the notation $\alpha !$ and $\partial ^{\alpha }$. Exactly as in the two variable case, it's proved by induction using integration by parts. The standard assumption is C^k. Hörmander is easy to follow, but the treatment for higher derivatives by the French school of Cartan, Dieudonné, Godement, Lang, et al, is probably too abstract. Even in Dieudonné's 1960 book, he assumes everything is going on in a Banach space. Hörmander does so, but in his case that there is no need. The results on symmetry of second derivatives and inverse function theorem in wikipedia are worked out versions of Hörmander. Showing that the inverse function theorem implies the implicit function theorem is easy (cf Lang, Analysis 1), just consider the mapping $(x,y)\mapsto (x,F(x,y))$. Conversely, the implicit function theorem implies the inverse function theorem: just consider $F(x,y)=f(x)-y$ (cf Dieudonné's Foundations of Analysis). The book of Krantz & Parks on the implicit function theorem covers the history of this. I hope this helps. Mathsci (talk) 02:12, 19 August 2020 (UTC)

It seems that what you say is already present here, even with a proof. Of course I have no problem with the idea of re-mentioning the statement of the theorem somewhere, but I'm confused by what purpose doing so does on this page. I feel the same about inverse function theorem and implicit function theorem, although the expositions on those pages could be strongly improved (for instance neither page seems to clarify that they're essentially identical results, as mathsci says). I don't plan on contributing to this page, just suggesting to consider two things: 1) consider improving or reformatting the individual topic pages so that it's easier to get whatever information one might need out of them (I think this is usually possible); 2) consider if there's any actual use to putting the results together. The current "Function theory in two variables" section is confusingly structured and unpleasant to read. This is because none of (a) mean value theorem, (b) Taylor's theorem, (c) the implicit and inverse function theorems, (d) the symmetry of second derivatives are particularly helpful for understanding one another. I can understand the use of a bulleted list of these results, linking to the individual pages, especially if it's a prelude to talking about calculus on manifolds. I just can't understand the use of the present morass of results on the page. Like I said, I don't plan to contribute to the page, do with it what you like Gumshoe2 (talk) 05:36, 19 August 2020 (UTC)

Hörmander's book start with "1.1 Review of Differential Calculus." It's pages 5-13. Apart from definitions the only non-trivial results are the inverse function theorem (Theorem 1.1.7) and the symmetry of second derivatives (Theorem 1.1.8). Readers of Hörmander's books will know that he often writes in a very condensed manner. The treatment of the two theorems here are now on wikipedia. (The appendix of Hitchin's notes on the inverse function theorem gives a similar treatment to that of Hörmander.) The equivalence of the inverse function theorem with the implicit function theorem is easy in the standard context. Gumshoe2 should feel free to add wiki content to implicit function theorem. On wikipedia the implicit function theorem is deduced from the inverse function theorem by the trick of Lang.

Parenthetic footmote: I noticed that in 2008 I added content about Matthias Günther's 1989 proof of the smooth Nash embedding theorem to wikipedia. Michael E. Taylor gives Günther's proof in section 14 of PDEs III. As Taylor comments, the proof relies on his "ingenious" use of the inverse function theorem for C¹ maps in a Banach space (page 90). The proof of the embedding theorem (pages 125-129) is first reduced to an n torus, by using the easy Whitney embedding theorem to embed a manifold in a box and hence an n torus. Instead of C^k norms, Sobolev norms can be used (usual definition with Fourier coefficients). The proof relies on applying the contraction mapping theorem. In this simplified version of the smooth embedding theorem, the proof of Nash-Moser inverse function theorem allows 100 pages to be reduced to about 2 or 3 pages. Mathsci (talk) 08:55, 19 August 2020 (UTC)

The response to @Gumshoe2:: you said "...confused by what purpose doing so does on this page". The point is not to state particular theorems but to give a *general ideal of consequence of differentiability*. This can be done in the form of Taylor's theorem or inverse function theorem, implicit function theorem, etc. By the way, there are also submersion theorem and constant rank theorem. Surely, it's more convenient if the readers didn't have to read all those separate articles. Like you I am not completely happy with the current presentation; I think we should emphasize the fact that "continuous" means approximately a constant function while "differentiable" means approximately a linear function. I mean, the definition of differentiability is precisely that a function approaches a linear function. This type of general idea is what needs to be emphasized; whether or how Taylor's theorem should be mentioned is just not an important question at all. -- Taku (talk) 01:54, 20 August 2020 (UTC)

Wikipedia is not a textbook!

This article is quite long, maybe in the top-1% of the longest math articles on Wikipedia, and contains a half-dozen or more sections with "this section needs expansion" templates. This seems wrong. Skimming what's written here, it seems to be an attempt to create a calculus textbook, hitting on all of the primary topics that one might see in multi-variate calculus. Wikipedia is not a textbook. There are existing wikipedia articles for all of the subtopics. I suggest content here could be (should be) moved to those articles, perhaps into a section that says "Example of what this looks like in Euclidean space". I mean, one of the coolest things about calculus is that it works so nicely in so many abstract spaces, as well as in engineering. For example, the upper-convected time derivative is the "Calculus on Euclidean space" version of the Lie derivative. But we wouldn't want to jam that into this article, any more than material derivative, which is also bog-standard "calculus on Euclidean space". These are central topics in Euclidean space, widely used in material science, yet oddly are absent in this article. By contrast, topics like weak derivatives are hugely important, but they've got approximately NOTHING AT ALL to do with Euclidean space. Weak derivatives aren't even "calculus"; they're a topic in analysis and Banach-space theory. Gauss-Bonnet is a high-importance/top-importance math topic, and its got more or less nothing at all to do with Euclidean space Its also not really "calculus", either; the honest label is to call it a topic in differential topology. This over-long article needs to be split into things that actually are important/central to Euclidean space, and to actual calculus, instead of a catch-all for generic math topics. 67.198.37.16 (talk) 21:40, 21 March 2024 (UTC)