A trivialization of a fiber bundle over a neighborhood 
is an isomorphism
between the restricted bundle and 
that commutes with the projection
to 
,
where 
is the fiber. For a vector
bundle, the isomorphism is linear
on each fiber. 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 
linearly 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.
A bundle can be constructed by gluing local products over a cover of the base manifold, using compatible transition functions between the trivializations.
