A chart tangent space is the tangent space at a point as represented in a coordinate
chart. The tangent space consists of all directions, or velocities,
a particle can take. In an open set 
in 
there are no constraints, so the tangent
space at a point 
is another copy of 
. The set 
could be a coordinate chart
for an 
-dimensional manifold.
The tangent space at 
, denoted 
, is the set of possible velocity
vectors of paths through 
. Hence the chosen coordinate
chart determines a vector basis: if 
are the coordinates,
then 
,
..., 
are a basis for the tangent
space, where 
is the velocity vector of a particle with unit
coordinate speed in the increasing coordinate 
direction.
The collection of all manifold tangent vectors
to every point on the manifold,
called the tangent bundle, is the phase
space of a single particle moving in the manifold 
.
It seems as if the tangent space at 
is the same as the tangent space
at all other points in the coordinate
chart 
.
However, while they do share the same dimension and
are isomorphic, in a change of coordinates,
they lose their canonical isomorphism.
For example, let 
and 
be coordinate ranges for the interior of the unit
interval 
.
We can change coordinates with 
defined by 
. This is a change of coordinates
because the derivative does not vanish on 
. But this change is not linear,
and stretches out 
more near 1 than it does near 0. The manifold
tangent vectors transform by the derivative. At
,
they are stretched by a factor of 
. At 
, they are stretched out by a factor
of 
.
In general, the manifold tangent vectors transform according to the Jacobian. The manifold
tangent vector 
at 
can also be considered as the manifold tangent
vector 
at 
in another coordinate chart, where 
is the diffeomorphism
from one coordinate chart to the other. The linear transformation determined by the Jacobian of 
is invertible, since 
is a diffeomorphism.
The Jacobian and the chain rule show that the tangent space is well-defined,
independent of coordinate chart, and also show
that manifold tangent vectors "push
forward." That is, given any smooth map 
between manifolds, it
makes sense to map the manifold
tangent vectors of 
to manifold tangent
vectors of 
.
Writing 
as the function 
between a coordinate chart
in 
and one in 
,
then 
maps 
from 
to 
. Another notation for
is 
,
the differential of 
.
In the language of tensors, these coordinate transformations
identify a vector field as a contravariant
tensor of rank 1.
A general smooth map pushes individual tangent vectors forward, but need not send a vector field
to a vector field on the target. If two points
have the same image, their pushed-forward vectors
need not agree. For a diffeomorphism 
, however, 
defines a vector
field 
on the target (Gualtieri 2018, p. 18).
