Over a small neighborhood 
of a manifold, a vector
bundle is spanned by the local sections defined on 
. For example, in a coordinate
chart 
with coordinates 
,
every smooth vector field can be written as a sum
where 
are smooth functions. The 
vector fields 
span the space of vector fields, considered
as a module over the ring of
smooth real-valued functions. On this coordinate
chart 
,
the tangent bundle can be written 
. This is a trivialization of the tangent bundle.
In general, a vector bundle of bundle rank 
is spanned locally by 
independent bundle sections.
Every point has a neighborhood 
and 
sections defined on 
, such that over every point in 
the fibers are spanned by those 
sections.
Similarly, for a fiber bundle, near every point 
, there is a neighborhood 
such that the bundle over 
is 
, where 
is the fiber.
A bundle is a set of trivializations that cover the base manifold. The trivializations are put together to form a bundle with its transition functions.
