Tangent space definition

Question

I am slightly confused by the algebraic definition of tangent spaces on a differentiable manifold. The book I am following defines the tangent space at a point p in a manifold M to be the set of all linear derivations on the algebra of germs of locally defined functions at p. What I do not understand is why we look at derivations on 'germs', i.e equivalence classes of functions whose restrictions agree on some open neighbourhood of p (explicated for my own edification), rather than just looking at derivations on the locally defined functions themselves. As I understand it, a derivation on the algebra of germs is the same as a derivation acting on representative members of the equivalence classes, so I do not understand why the object of consideration is the set of equivalence classes of functions rather than the functions themselves. The wikipedia article on tangent spaces briefly justifies this by asserting that this is necessary for some applications in Algebraic Geometry, but I did not follow any of this.

Answer 1

Suppose $U$ and $V$ are two neighborhoods of $p$, $f$ is defined on $U$ and $g$ is defined on $V$. How do you define $f+g$? (You do need at least an abelian group structure if you want to define derivations.)

What you suggested in the comments is the logical solution: take $f+g$ defined on $U\cap V$. But then if I define $f_2$ as the restriction of $f$ on $U\cap V$ (which is a different funtion from $f$!), then $f_2+g$ is also $f+g$ on $U\cap V$. So $f_2+g=f+g$, and $f_2=f$. Ooops.

The solution to that is to say that for our purposes, $f$ and $f_2$ should actually be considered the same function, because we are only interested in what happens near $p$. We don't care if $f$ is actually defined on a bigger neighborhood.

But this is exactly the definition of germs: we identify functions that coincide on some smaller neighborhood. You see that this is necessary at the very least for the sum of functions to be well-defined.