1. THE STOKES THEOREM AND THE DE RHAM THEOREM

1.1. The Stokes Theorem. Let M be a C ∞ manifold of dimension m. Let U ⊆ M be an open subset of M. We indicate by Ap(U) the vector space of complex-valued p-forms on U. More precisely, if TM is the tangent bundle of M and TCM: = TM ⊗R C is the complexified tangent bundle of M, a complex-valued p-form on U is a C ∞ section of the vector bundle Λp(TCM) ∗ on U. If U is a coordinate set with {x1,..., xm} local coordinates then a C ∞ section ω of Λp(TCM)∗ on U is given by ω = 1≤i1<...<ip≤m fi1...ip(x)dxi1 ∧... ∧ dxip, for some complex-valued C ∞ functions fi1...ip defined on U. A set R ⊂ M is a manifold of dimension m with C ∞ boundary if IntR is a m-dimensional manifold and for any p ∈ ∂R there exists a open coordinate set U in M with local coordinates {x1,..., xm} such that R ∩ U = {q ∈ U |x1(q) ≤ 0} and ∂R ∩ U = {q ∈ U |x1(q) = 0}. Thus ∂R is a (m − 1)-dimensional manifold. Moreover if M is oriented and the above system of local coordinates {x1,..., xm} is positive, we give ∂R the orientation coming from declarin