Hello,

We have defined the tangent space at a point p of a manifold as the set of all derivatives.

Now i want to show, that there exist a bijection between all derivatives and the "Tangentvectors" defined by (equivalent classes of) paths.

More precisely:

We call two paths equivalent, iff they define the same derivation at a point p. Now i want to show that there is a bijection between the set of all equivalent classes and the Tangentspace (defined as the set of derivatives).

My Approach was the following.

Every path c:I->M into the manifold M defines a derivation by

$\displaystyle D(f):=\frac{d}{dt}_{|t=0}(f\circ c)(t)$

so i try to show that $\displaystyle \phi([c])=D$ with $\displaystyle D(f)=\frac{d}{dt}_{|t=0}(f\circ c)(t)$ is a bijection.

Of course $\displaystyle \phi$ is well defined and injective, but i couldn't show that

$\displaystyle \phi$ is surjective!

If D is some arbitrary derivative at the point p. How can i find a path c, s.t. $\displaystyle \phi(c)=D$?

Regards