  Differential forms on the Fr\'echet manifold F(S,M) of smooth functions on a
compact k-dimensional manifold S can be obtained in a natural way from pairs of
differential forms on M and S by the hat pairing. Special cases are the
transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat
pairing with a constant function) and the bar map associating p-forms on F(S,M)
to p-forms on M (hat pairing with a volume form). We develop a hat calculus
similar to the tilda calculus for non-linear Grassmannians.
