The tangent bundle 
of a smooth manifold 
consists of its tangent spaces 
at all points 
. Its elements
are tangent vectors together with their
base points:
 |
(1)
|
The tangent bundle is a vector bundle of bundle rank 
,
where 
is the dimension of 
. A coordinate chart on
provides a trivialization.
In coordinates 
, the vector fields 
form a vector
basis of the tangent space at every point
in the coordinate chart. The transition
function between two such trivializations
is the Jacobian of the change of coordinates.
For example, near the point 
on the unit sphere, consider
two coordinate charts parametrizing the same
hemisphere, 
and 
:
 |
(2)
|
 |
(3)
|
with 
and 
.
The change of coordinates is 
:
 |
(4)
|
Its Jacobian is
![[cosx_1cosx_2 -sinx_1sinx_2; 0 cosx_2]](/images/equations/TangentBundle/NumberedEquation5.svg) |
(5)
|
with determinant 
, so it is invertible
on 
.
The coordinates of a tangent vector transform by this Jacobian. If the vector
has coordinates 
in the first coordinate chart,
then its coordinates in the second coordinate
chart are 
at 
.
Both descriptions represent the same element of the tangent
bundle.
