Smooth vector bundles and local coordinates

In this entry we remind the reader of the definition of a smooth vector bundle and give a local coordinates for a smooth section. 

Definition [Milnor-Stasheff] A rank n vector bundle is a map and a vector space structure on for all so that and fiber-preserving homeo so that is a v.s. iso.  To define a smooth vector bundle one requires that E and X are manifolds and the ‘s are diffeomorphisms. 

Definition [Steenrod (see also Davis-Kirk)] A rank n vector bundle is a map and a collection of homeos so that

each is fiber-preserving over . 

is an open cover of

so that

is max’l with respect to the above three properties. 

To define a smooth vector bundle one requires that X is a manifold and the are smooth. 

A smooth section is a smooth map s.t. . 

The vector space of smooth sections is a module over the ring of continuous functions . 
e.g.

A smooth section of an -plane bundle over an -manifold is locally an element of .  To make this precise and charts and .  Then via