example for dimensions. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains. It thus suffices to prove Stokes' theorem for sufficiently fine tilings (or, equivalently, simplices), which usually is not difficult. Classical vector analysis example Let be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that divides into two components, a compact one and another that is non-compact. Let denote the compact part that is bounded by and suppose is smooth, with . If is the space curve defined by and is a smooth vector field on , then: This classical statement, is a special case of the general formulation after making an identification of vector field with a 1-form and its curl with a two form through Generalization to rough sets The formulation above, in which is a smooth manifold with boundary, does not suffice in many applications. For example, if the domain of integration is defined as the plane region between two -coordinates and the graphs of two functions, it will often happen that the domain has corners. In such a case, the corner points mean that is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above does not apply. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. This is because and its boundary are well-behaved away from a small set of points (a measure zero set). A version of Stokes' theorem that allows for roughness was proved by Whitney. Assume that is a connected bounded open subset of . Call a standard domain if it satisfies the following property: there exists a subset of , open in , whose complement in has Hausdorff -measure zero; and such that every point of has a generalized normal vector. This is a vector such that, if a coordinate system is chosen so that is the first basis vector, then, in an open neighborhood around , there exists a smooth function such that is the graph and is the region . Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff -measure and a finite