From the point of view of coordinate charts, the notion of tangent space is quite simple. 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 there is a canonical vector
basis: if 
are the coordinates, then 
are a basis for the tangent space, where 
is the velocity vector of a particle with unit speed moving
inward along the coordinate 
. The collection of all 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
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 charts for 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 tangent vectors transform
by the derivative. At 
, they are stretched by a factor of 
. While at 
, they are stretched out by a factor of 
.
In general, the tangent vectors transform according to the Jacobian. The tangent vector 
at 
can also be considered as the tangent vector 
at 
in another coordinate chart, where 
is the diffeomorphism
from one chart to the other. The linear transformation determined by the Jacobian
of 
is invertible, since 
is a diffeomorphism.
Not only does the Jacobian, and the chain rule, show that the tangent space is well-defined,
independent of coordinate chart, but it also shows that tangent vectors "push
forward." That is, given any smooth map 
between manifolds, it makes sense to map the tangent
vectors of 
to 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,
the tangent vector's pushing forward means that a vector field is a covariant
tensor.
