The cotangent space 
at a point 
of a smooth manifold 
is the dual vector space
of the tangent space 
. Its elements are the cotangent
vectors, or covectors, at 
, namely the linear functionals
 |
If 
has dimension 
, then so does 
. A coordinate chart
with local coordinates 
,
..., 
gives the basis 
, ..., 
of 
. The union of the cotangent
spaces over all points 
is the cotangent bundle 
.
