690:, 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
363:" 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, Knowledge 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. --
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treatment here will be more concise. Itâs 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; thatâs not necessary bad. Taylorâs 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. â-
834:. 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. --
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866:, 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
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of the primary topics that one might see in multi-variate calculus. Knowledge 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
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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. --
712:(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
531:. Exactly as in the two variable case, it's proved by induction using integration by parts. The standard assumption is C. 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
359:, 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 "
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826:: 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
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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
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This article is quite long, maybe in the top-1% of the longest math articles on
Knowledge, 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
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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
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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,
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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.
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In my opinion itâd be much easier to justify an article on calculus on manifolds in
Euclidean space, as in Spivakâs book. Anyway, it might be helpful to be specific. Whatâs to be gained from discussing Taylorâs theorem here as opposed to on its own page?
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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. --
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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:
682:, 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
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Sure, I understand the idea. But what would you say about Taylorâs theorem here that you couldnât easily say on its own page? French history is quite complex; I think Taylorâs theorem isnât!
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to wikipedia. Michael E. Taylor gives GĂŒnther's proof in section 14 of PDEs III. As Taylor comments, the proof relies on his "ingenious" use of the inverse function theorem for C maps in a
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302:(Guillemin & Pollack), way beyond elementary calculus. It's quite unrealistic to write a low level article that combines both elementary calculus and differential topology.
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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. --
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664:(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.
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Hörmander's book start with "1.1 Review of
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are also on wikipedia (I inserted one); the implicit function theorem is a corollary. Most of the proofs have been taken from Hörmander.
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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
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As I've written to Taku, his suggestion on calculus was a very idea and the skeleton he suggested is excellent.
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Parenthetic footmote: I noticed that in 2008 I added content about
Matthias GĂŒnther's 1989 proof of the smooth
716:. On wikipedia the implicit function theorem is deduced from the inverse function theorem by the trick of Lang.
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It seems that the current content in the draft could very easily and naturally be included in other articles.
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in wikipedia are worked out versions of Hörmander. Showing that the inverse function theorem implies the
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I replied "Seems like a good idea, if it that content doesn't already exist on wikipedia. The case of
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Calculus is a fairly elementary topic. The other topics are of a different flavour, topics from
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which seems to me to be synonymous with calculus on Euclidean space
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