Every smooth manifold
has a tangent bundle ,
which consists of the tangent space at all points in . Since a tangent space is the set of all tangent vectors to at , the tangent bundle is the collection of all tangent vectors,
along with the information of the point to which they are tangent.

(1)

The tangent bundle is a special case of a vector bundle. As a bundle it has bundle rank , where is the dimension of . A coordinate chart on
provides a trivialization
for .
In the coordinates, ),
the vector fields ,
where ,
span the tangent vectors at every point (in the coordinate
chart). The transition function from these coordinates to another set of coordinates
is given by the Jacobian of the coordinate change.

For example, on the unit sphere, at the point there are two different coordinate
charts defined on the same hemisphere, and ,

(2)

(3)

with
and .
The map between the coordinate charts is .

(4)

The Jacobian of is given by the matrix-valued function

The tangent vectors transform by the Jacobian. At the point in , a tangent vector corresponds to the tangent vector at in . These two are just different versions of the same element
of the tangent bundle.