Tangent vectors on manifold, different definitions 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.

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.    
Notes
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...
