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 (Gualtieri 2018, pp. 14-16).
For example, near the point 
on the unit sphere, consider
two parameterizations of 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.
