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I'm not sure what is being asked here. Does $\mathbb{Z}[x_1,x_2,\dots,x_{n+1}], \prod x_i = 1$ refer to $\mathbb{Z}[x_1,x_2,\dots,x_{n+1}]/(x_1 x_2 \cdots x_{n+1} - 1)$, more commonly known as the ring of Laurent polynomials $\mathbb{Z}[x_1,x_1^{-1},\dots,x_n,x_n^{-1}]$?
Also, by asking permission, you let the author know that their notes are useful to others. It gives you the opportunity to thank them for making the notes available and will likely make them more enthused about adding new notes, updating notes, etc. in the future. It could also potentially lead to the author seeking your feedback and possibly incorporating some of your suggestions into a revised version down the road. So even if asking permission is a formality in a given situation, it can have desirable side effects.
And you can always briefly explain the relevant formula if it is something you suspect they might not know. I find it important to make sure to introduce the relevant background as part of the problem (as opposed to explaining while you go through the solution), to make it clear that you don't necessarily expect students to know the relevant physical law (though it's great if they do!). And it is certainly worth telling students that the more physics, chemistry, biology, economics, etc. that they learn, the more applications of calculus they will discover.
@Keith: I like your example. I had in mind showing that the naive product rule fails by the following argument: Knowing how $P$ and $V$ relate (obvious from $PV = C$, requires no calculus as you point out), we see that $dP/dt \cdot dV/dt$ is always a non-positive number, and is negative if both $P$ and $V$ are not static; on the other hand $d{PV}/dt$ is always 0. I find this convinces many students of "why" the actual product rule has to be used. Of course, really it only proves why the naive product rule fails! So I do try to point this out as well, though students often miss this point.
I like Boyle's Law and total resistance of two resistors in parallel. Both are found in almost any textbook. The value of Boyle's Law examples in particular is that simple related rates calculations confirm something they already know (increasing pressure of a gas decreases its volume and vice versa). There's a lot of value of relating a new concept you are learning to things you already know, though most students fail to grasp this metaprinciple.
@darij: I generally agree with your perspective. I think the actual form of Jordan normalization is less important than the question that it answers. Ideally, a student who takes courses in linear algebra and abstract algebra would not come away with the idea that the subjects have nothing in common (other than the word 'algebra' in their name) -- but my suspicion is that most students come away with just that conclusion.
On second thought, proving that your variety is irreducible is most easily done by showing its an orbit variety, so I retract my previous approach. Instead, fix an $n+1$-diml v.s. $V$ and let $W_1, W_2$ be subspaces of dimension $m+1$ and $n-m+1$ that intersect in a $1$-diml subspace. You can find a basis, call it $\mathscr{V} = \{v_0, \dots v_n\}$ as in the above comment so that $W_1$ and $W_2$ have the form of the $H$ and $H'$ above. Then change of basis from $\mathscr{E}$ to $\mathscr{V}$ gives you an element of $PGL_{n+1}$ that takes one pair of such subspaces to the other.
