Exterior derivative
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Let 
be a smooth function on a manifold. The differential or exterior derivative 
is a covector field on M defined as follows: for v a tangent vector at a point 
i.e., 
is the directional derivative of f in the direction v.
Note that if 
are a local coordinate system for M at p, then 
define a local co-frame near p. Thus, near p, we may write the differential of f as a linear combination:
In fact, since 
, we get that:
