Over a small neighborhood U of a manifold, a vector bundle is spanned by the local sections defined on U. For example, in a coordinate chart U with coordinates (x_1, ..., x_n), every smooth vector field can be written as a sum sum_i f_i d/dx_i where f_i are smooth functions. The n vector fields d/dx_i span the space of vector fields, considered as a module over the ring of smooth real-valued functions. On this coordinate chart U, the tangent bundle can be written U×R^n. This is a trivialization of the tangent bundle.

bundle | fiber bundle | manifold | transition function | vector bundle