Germs: Why is it sensible to define a function on a collection of equivalence classes by its action on each element?

Question

I am following Loring W. Tu in his second edition of 'An introduction to manifolds'. Here is a pdf-copy of the book. On page 87 he defines $C^\infty_p(M)$ as the set of germs of $C^\infty$-functions at $p\in M$. He then defines a derivation as a map $D : C^\infty_p(M) \to \mathbb{R}$. In contrast, half way down the page he defines the partial derivative in coordinates as a map $\partial/\partial x^i|_p : C^\infty(M) \to \mathbb{R}$, and claims that it is easy to check that this is a derivation.

I do not understand this because $\partial/\partial x^i|_p$ cannot be a derivation when it's domain is $C^\infty(M)$ and not $C^\infty_p(M)$. He also casually seems to talk of functions as if they are equivalence classes. To me it seems too simple that the author has mistaken the set of equivalence classes for its elements. Is there a principle regarding equivalence classes or germs that are being applied implicitly that I am missing? Could you explain please?

Answer 1

In general if $\mathcal A$ is a commutative algebra and $M$ is an $\mathcal A$-module, we will say that a linear map $D : \ \mathcal A \to M$ is a derivation if $D(fg)=f Dg + g Df $ for $f, g \in \mathcal A$. Operation $\left. \frac{\partial}{\partial x^i} \right|_p$ satisfies this property, no matter if we consider $C^{\infty}(M)$ or the space of germs as its domain.

Secondly, given an element of $C^{\infty}(M)$ we can always consider its germ at $p$. Even though partial derivative at $p$ was initially defined on functions on $M$, its value actually depends only on the germ at $p$. Moreover given any germ at $p$ you can always extend it to a smooth function on whole $M$ using bump functions, so in the end when evaluating derivatives of any finite order at a point it doesn't make much difference if you work with functions on $M$, functions on some fixed neighbourhood of $p$ or germs at $p$.

Answer 2

The key idea is that germs of functions are those on which the partials agree, no matter which function from the germ you choose.

Let me give a very concrete example: we can say that for functions from the real to the reals, there's an equivalence relation: $f$ and $g$ are equivalent if and only if $f(0) = g(0)$.

Now I define the "zero-square" $Z(f)$ of a function $f$ to be $f(0)^2$. Then I claim that a very slight generalization of "zero-square" makes sense as a function on equivalence classes. Why? Because if $f\sim g$, then $Z(f) = f(0)^2 = g(0)^2= Z(g)$, so $Z$, evaluated on ANY member of an equivalence class, gets you the same number. We say that this slight generalization (which usually is given some typographically similar name, like $\bar{Z}$, but is sometimes just denoted with the same letter (alas) is "well defined on equivalence classes".

That's what Tu is doing here. The partial-derivative-at-$p$ map acts on all $C^\infty$ functions defined in a neighborhood of $p$; for any two elements of the same germ at $p$, you get the same result. Hence the partial-derivative-at-$p$ map ends up being "well defined on germs".

You're right that this is, to some degree, an abomination -- it's yet another of those mathematical puns, like writing $dz/dx = dz/du \cdot du/dx$, where the $z$ on the left and the $z$ on the right are two completely different functions. But it's also really commonplace...

Answer 3

Tu could have mentioned the following definition in Section 8.1:

$$D\left([f]_p \right) := D(f) \equiv Df,$$

where $[f]_p \equiv [(f, U)]$ is the equivalence class of $(f, U)$. Here $f: U \to \mathbb{R}$ is a smooth function on $U \ni p \in M$. Although he mentions this fact in Section 8.2, right after Eq. (8.1).

To show that the partial derivatives at point $p$ is a derivation at $p$, one needs to consider the algebra structure of $C^{\infty}_p(M)$, which is partially discussed here.

This issue has been discussed in a number of questions on this site. I list some of them below.

1. Derivation at a point factors through derivation on algebra of germs
2. Germs of a $C^{\infty}$ function at $p$, and the derivation at $p$.
3. different definitions for $C^{\infty}(p)$