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
![[cosx_1cosx_2 -sinx_1sinx_2; 0 cosx_2]](/images/equations/TangentBundle/NumberedEquation5.svg) |
(5)
|
which has determinant 
and so is invertible on 
.
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.
