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  1. 'Marginally generalized theorems' refers to theorems that were generalized so marginally that they can still be taught, and understood by the students, in the same course.

Examples of theorems NOT marginally generalized:

  1. A skilled high-schooler can understand Puiseux series, Dirichlet's theorem on arithmetic progressions, and Lagrange's four-square theorem. But their generalizations are too cavernous.

Examples of theorems that can be marginally generalized:

  1. Spivak proves $\sqrt{2}$'s irrationality fully, but banishes $a^{\frac 1b}$'s to the exercises. Isn't proving (only) the latter more efficient?

  2. Why don't multivariate calculus textbooks introduce Stokes's Theorem (with a double integral, not differential forms) first, and then state Green's as a corollary? Calculus: Early Transcendentals (6 edn 2007) introduces the latter as §16.4, and former as §16.8.

  3. Why not prove first Gauss's generalization of Wilson's Theorem:

$\forall \; p \; \text{odd prime}, \alpha \in \mathbb{N} : \quad \prod_{k = 1 \atop \gcd(k,m)=1}^{m} \!\!k \ \equiv \begin{cases} -1 \pmod{m} & \text{if } m=4,\;p^\alpha,\;2p^\alpha \\ \;\;\,1 \pmod{m} & \text{otherwise} \end{cases}?$

I'm not referring to the further generalization of Wilson's Theorem requiring group theory.

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  • $\begingroup$ Let's consider your example of generalizations of Wilson's theorem. It's difficult to export the group-theoretic insights to more elementary contexts. For example, see this MSE answer where I struggled to do so (but I don't think I succeeded). Is that worth the effort in elementary textbooks? Perhaps only so if you can reuse some of the ideas later in the course. Otherwise it may be better to simply mention the generalization (to motivate further study), and save the space for proofs whose concepts can be better understood at that level. $\endgroup$ – Bill Dubuque May 18 '18 at 19:42
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    $\begingroup$ Why? Define the benefit. Define the cost. Compare the two. $\endgroup$ – guest May 18 '18 at 20:28
  • $\begingroup$ @guest Sorry. I don't understand. Consider 1. Proving $a^{1/b}$ irrational first would save cost and space. No need to write 5 exercises on $\sqrt{2}, \sqrt{3}, ...$ is irrational. $\endgroup$ – Greek - Area 51 Proposal May 19 '18 at 1:31
  • $\begingroup$ @Number I agree, but I didn't intend to refer to group-theoretic proofs of number theory theorems though. That Gauss's generalization doesn't require group theory, or does it? If yes, I'll delete it and instance another theorem. $\endgroup$ – Greek - Area 51 Proposal May 19 '18 at 1:32
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    $\begingroup$ @Greek But I don't (explicitly) use group theory. Rather, I strive to present the proofs in a more elementary language that remains faithful to the group-theoretic viewpoint. My point is that a good teacher will strive to do the same, i.e. strive to highlight the essence of the matter. The more you generalize the more difficult this task becomes - both for the teacher and the students. One of the qualities of a great teacher is that they have sufficient mastery of the subject that they can quickly ascertain the optimal level of generality appropriate for the targeted audience. $\endgroup$ – Bill Dubuque May 19 '18 at 2:00
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Spivak proves $\sqrt{2}$'s irrationality fully, but banishes $a^{1/b}$'s to the exercises. Isn't proving (only) the latter more efficient?

Yes, it's certainly more efficient. So what? Efficiency is not a primary goal of a textbook.

Generality often makes concepts harder to understand. That's true for professional mathematicians, and it's even more true for students, who have less experience doing the task of concretizing ideas on their own.

The proof that $\sqrt{2}$ is irrational adequately conveys the idea of the proof - someone who has truly understood it will be able to generalize it to $a^{1/b}$ without much difficulty. But many students will find the idea easier to understand the first time they see it by looking at a specific number like $2$ rather than an abstract number $a$.

As a side benefit, it gives students the opportunity to see whether they understand the argument by doing the generalization themselves.

Most multivariate calculus textbooks devote 1 chapter to Green's Theorem. Why not introduce Stokes's first, and then state Green's as a corollary?

Because Stokes' Theorem is too abstract for many students at that level to understand in a single go. Green's Theorem is a warm-up, getting students more used to the idea, before they encounter the more general theorem.

Abstraction imposes cognitive work on the learner, to think through the scope of the abstraction and to produce concrete examples themselves. A good textbook has to balance that work against other important cognitive work by the learner, like thinking about the intuition and proof of a theorem.

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  • $\begingroup$ Your penultimate para.: I clarified my intended signification of Stokes's Theorem. $\endgroup$ – Greek - Area 51 Proposal May 19 '18 at 5:14
  • $\begingroup$ @Greek-Area51Proposal: I was referring to the double integral version of Stokes' theorem. $\endgroup$ – Henry Towsner May 19 '18 at 13:07
  • $\begingroup$ Thanks. OK. But how is Stokes's " too abstract for many students at that level to understand in a single go"? Please see my reference to Stewart in my edited para. 4. $\endgroup$ – Greek - Area 51 Proposal May 19 '18 at 17:59
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    $\begingroup$ @Greek-Area51Proposal It's a statement about 3D objects (which are harder to draw or visualize) involving surface integrals (which are more abstract than double integrals). $\endgroup$ – Henry Towsner May 19 '18 at 19:13
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    $\begingroup$ @Greek-Area51Proposal: No, I haven't reviewed Stewart's exposition (but, based on the table of contents, it's similar to Thomas'). I'm not arguing that presenting Green's first is the only conceivable choice, only that there are good reasons to consider it (which might have to compete with others). The drawback to presenting Stokes' first and using Green's as an illustration is that it's asking students to simultaneously understand Green's and to understand how Green's illustrates Stokes'; understanding Green's is hard enough as is (Stewart devotes an entire section to it). $\endgroup$ – Henry Towsner May 20 '18 at 0:52
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I'll answer the question with respect to the Green's/Stokes' theorem special case. The summary is that, from a pedagogical point of view, little is gained by passing directly to the general case, and much is potentially lost (students for example).

If by Stokes' theorem is meant the version relating surface integrals and line integrals, this theorem is substantially harder to teach than its planar special case usually known as Green's theorem. Green's theorem relates an ordinary double integral over a region with a line integrals over the region's boundary. The version for surfaces requires defining surface integrals, and this introduces substantial new difficulties. In particular students have a lot of trouble with parameterizing surfaces, calculating normal vectors, and understanding orientations of surfaces. With Green's theorem only the orientation issue is present as such, and it is more easily visualized in the planar context, where it already is difficult for many students. In any case, objects in three dimensions are almost always harder to understand than their counterparts in two dimensions, if only because it is easier to draw pictures of planar objects.

If by Stokes' theorem one means the full blown theorem in terms of differential forms, then the difficulty, both technically and conceptually, is substantially greater. Moreover, the theorem is usually formulated in terms of differential forms (why stop there? one could go on to currents and just teach out of Federer ...). Understanding differential forms requires understanding duality of vector spaces, a topic often omitted from introductory linear algebra courses, often omitted completely from instruction given to physicists and engineers, and itself requiring some effort to assimilate properly. Many professional engineers never learn that the curl of a vector field is the vector dual to the exterior differential of the one-form dual to the vector field; physicists mostly can interpret such a phrase, but even applied physicists (as opposed to high energy theoreticians or general relativists) don't usually think that way, and don't write Maxwell's equations using forms (obviously theoreticians do).

While in fact the general theorem is not much more difficult to prove rigorously than are its various special cases (although to do so in full generality one has to deal with triangulations, or partitions of unity, or something along those lines) doing so obscures the essential ideas, which are just as well understood in proving special cases such as Green's or Gauss's (the divergence) theorem. In fact, once one understands the proof of Green's theorem, one basically understands what is needed to prove the general theorem; all the essential issues are present already.

Very few students learn top down (only rarely does one have a future Grothendieck as a first-year student). Most learn best proceeding from the special to the general. One pares away as many inessential aspects as one can, to focus the student's attention on the core issues.

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  • $\begingroup$ Your para. 2: But multivariate calculus books cover them both in the same chapter? Your para. 3: No. I didn't intend to refer to the differential form generalization. $\endgroup$ – Greek - Area 51 Proposal May 19 '18 at 5:14
  • $\begingroup$ @Greek-Area51Proposal You mean, "your second paragraph" - since the paragraphs are not numbered :) $\endgroup$ – fabspro May 19 '18 at 14:05
  • $\begingroup$ As an aside, understanding differential forms doesn't require "understanding duality of vector spaces". While that's one path for exposition, it is by no means the only one. In the context of multivariable calculus, all you need is the idea of distinguishing between row and column vectors! Heck, I came up with that idea on my own without having any clue about differential geometry, simply because it made Calc III easier for me to understand (while I admittedly knew linear algebra at the time, I didn't know anything about dual spaces). $\endgroup$ – user797 May 19 '18 at 14:17
  • $\begingroup$ ... if there is an actual technical obstacle to introducing differential forms early on in calculus, it's about going to higher tensor rank: i.e. 2-forms, 3-forms, and so forth. It's not obvious to me that this is an essential obstacle, though. $\endgroup$ – user797 May 19 '18 at 14:19
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    $\begingroup$ @Hurkyl: "distinguishing between row and column vectors" is understanding duality. I'm talking about duality of vector spaces, not something deeper. It's not an issue of exposition, it's an issue of conceptual clarity. A one-form is an element in the dual vector space, a differential one-form is a smooth section of the cotangent bundle, not the tangent bundle. A lot of (completely unneeded) confusion is generated by expositions that identify one-forms and vectors without being explicit about doing so! $\endgroup$ – Dan Fox May 19 '18 at 15:26
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I would like to add that progressing from the special to the general is a good way to teach how research actually works. Frequently you discover some relation by accident and only after working with that relation for some time you realize that there could be some generalization for it. By using intuition and experience you are probably able to extract what is common to all the special cases you have explored so far and finally derive the general theorem.

The textbooks already make things more 'efficient' in that they reveal to the reader that the general theorem indeed exists which is usually already a great leap/shortcut for the learner.

But I understand very well the desire to express things more compactly than the textbooks do. I have always done this for myself, just to discover in the end, that my 'summaries' can only be understood by myself and because I had read the original textbook before.

Although reading a textbook takes a lot more time, I find it also a lot more enjoyable than reading a formulary for example, just because a (good) textbook gives me the feeling as if I were taking part in the original discovery of the facts it presents. And joy is certainly the number one success factor for learning.

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