# Why don't textbooks foreground marginally generalized theorems? [closed]

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:

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

• 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. – Tobias Kildetoft May 17 '18 at 7:20
• @TobiasKildetoft +1. Thanks. I corrected my example. – NNOX Apps May 17 '18 at 7:29
• This might be a pedagogical question rather than a mathematical one. – Teddy38 May 17 '18 at 7:40
• 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. – Mark Bennet May 17 '18 at 7:41
• Asked again at Mathematics Educators Stack Exchange. – Joel Reyes Noche May 28 '19 at 8:14

## 1 Answer

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.