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A differential 1-form is a covector field: it associates to every pt p a covector in the dual space of TpM. Given a smooth function f, one can define the differential 1-form df which is basically (-,f), i.e. If X is a vector field then at p, df(X) = (X,f). Recall for smooth f: X to R we have the notion of differential f_* at p which is a map between tangent space. If we identify T_f(p) R with R then f_* and df at p are the same map.
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