The classical calculus of variation is a critical point theory of certain differentiable functions (or functional) on a smooth or piecewise smooth path space, whose differentiable structure is defined implicitly. Because of the importance of path spaces to analysis, geometry and other fields, it is desirable to develop a geometric integration theory or a de Rham theory for path spaces. Having in mind this general goal, we are going to consider a large class of path space differential forms, which can be constructed from usual differential forms by a method of iterated integration.Recall that the Poincaré lemma is proved through a process of integration, which converts every closed /?-form w on a manifold M locally to a (p -1)-form. The same process can be used to obtain a (p -l)-form fw defined globally on the total smooth space P(M) of M. More generally, given forms w v . . ., w r on M y we may repeat the integration process r times in an appropriate manner and obtain a differential form fw { • • • w r on P(M) of degree -r + 2 1

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