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. 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?

  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 refer not the further generalization requiring group theory.

  • 3
    $\begingroup$ The last one: Because the generalized version requires the students to know what a group is, whereas Wilson's theorem is purely about modular arithmetic. $\endgroup$ – Tobias Kildetoft May 17 '18 at 7:20
  • $\begingroup$ @TobiasKildetoft +1. Thanks. I corrected my example. $\endgroup$ – NNOX Apps May 17 '18 at 7:29
  • $\begingroup$ This might be a pedagogical question rather than a mathematical one. $\endgroup$ – Teddy38 May 17 '18 at 7:40
  • $\begingroup$ Suggestion: eg on 3 - because it is sometimes best to teach a method by using a concrete example which can be easily motivated/understood. Then abstract/generalise when you have a model to follow. And note also this generalises further via the rational root theorem. $\endgroup$ – Mark Bennet May 17 '18 at 7:41
  • $\begingroup$ Asked again at Mathematics Educators Stack Exchange. $\endgroup$ – Joel Reyes Noche May 28 '19 at 8:14

I don't have enough reputation to post this as a comment but a very standard proof of Stokes theorem in 3 dimensions is to reduce the problem and apply Green's theorem.


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