A section of a fiber bundle gives an element of the fiber over every point in
. Usually it is
described as a map 
such that 
is the
identity on 
. A real-valued function on a manifold
is a section of the trivial line
bundle 
. Another common example is a
vector field, which is a section of the tangent
bundle.
The rank of a vector bundle is the dimension of its fiber. Equivalently, it is the maximum number of linearly independent local bundle sections in a trivialization. Naturally, the dimension here is measured in the appropriate category. For instance, a real line bundle has fibers isomorphic with
, and a complex
line bundle has fibers isomorphic to 
, but in both cases
their rank is 1.
The rank of the tangent bundle of a real manifold
is equal to the dimension of 
. The rank of a
trivial bundle 
is equal to 
. There is no upper
bound to the rank of a vector bundle over a fixed manifold 
.Source : WMW