Hi, I'm working with different definitions of tangent vectors and trying to show they are equivalent. Here

is a smooth manifold of dimension

and

is the set of smooth, real valued functions defined on an open neighbourhood of

.

Definition 1: A tangent vector at

is the tangent vector to some curve

, i.e. a map

defined by

.

Definition 2: A tangent vector at

is a linear functional

satisfying the Leibnitz product rule:

for all

.

Definition 1

Definition 2 is obvious. To show Definition 2

Definition 1, let

be a chart around

with coordinate functions

. Define

so that

.

Now define a curve

so that, for any

,

. Hence

is the corresponding curve to

.

I have two questions. First of all: is this correct? Secondly: if so, then why have we not used the Leibnitz product rule (and so could this not be true for linear functionals

not satisfying the product rule)?

Any help would be appreciated.