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.