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I have problem with understanding difference between definitions of tangent vectors. Can you explain which definition of tangent vector is the best? (in case of finite dimensional manifold) I know 3 definitions, germs, curves, operators (they are equivalent). Is there some another? How in these definitions we use partition of unity? How we use jets in manifold theory? Thank you for your answers.

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Tangent vectors are not germs of anything. – Georges Elencwajg Nov 16 '13 at 23:12
@GeorgesElencwajg They can be defined as equivalence classes of germs at the point in question. See, for example, these notes. – zibadawa timmy Nov 16 '13 at 23:14
As far as which is "best", that depends entirely on the situation at hand. Sometimes a theorem is easy to prove with one, and hard or nearly impossible with another. Chain rule using germs is really easy. Establishing vector space structure with germs is a little harder than you might like. In that sense, they are all the best, as it is important to know all of them so that you can apply whichever one will suit the situation best. – zibadawa timmy Nov 16 '13 at 23:17
@zibadawa timmy: An equivalence class of germs is not a germ. By the way, who wrote the notes you link to? – Georges Elencwajg Nov 16 '13 at 23:27
Yes, there is another definition: see my answer. Partitions of unity are irrelevant in the definition of tangent vectors. – Georges Elencwajg Nov 16 '13 at 23:59

The oldest way to define a tangent vector to a manifold is the one the physicists devised long before mathematicians (essentially Hermann Weyl and Hassler Whitney) came up with their rigorous definition of a manifold.
It is as follows:

A tangent vector to the manifold $M$ at $m\in M$ is a map $c$ attaching to each chart $\phi:U\to U'\subset \mathbb R^n$ of $M$ with $m\in U$ a vector $c(\phi)\in \mathbb R^n$ with the condition that for any other chart $\psi:V\to V'\subset \mathbb R^n$ the equality $c(\psi)=D_{\phi(m)}(\psi\circ \phi^{-1})\cdot c(\phi)$ obtains.

1) The domain of the map $c$ is the atlas of $M$, a huge set!, and its codomain is $\mathbb R^n$.

2) Physicists actually write this in a sloppier but much clearer way: they say tangent vectors transform in a contravariant way and write cute upper indices instead of composition of maps...

3) I learned this definition in Spivak, Dieudonné and physics books long ago...

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