By George Arnsby-Jones
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A topological space homeomorphic to a simpliceal complex is called triangulated. In the following we work only on these triangulated spaces. Based on the triangulation K of a given manifold we can construct the Abelian groups Cp (K), p = 0, . . , n freely generated by the oriented p-simplexes of K, with integer coefficients, called the chain group. 1) with the action ∂p σ p = j=0 (−1)j [v 0 , . . , v j−1 , v j+1 . . , v p ] creating thus a (p − 1)-simplex. 2) which is the central property of homology, and somehow the main philosophy of the compact surfaces, contours, boundaries in general: The boundary of a boundary is the empty set.
We can generalize the integrability concept for a general manifold. Definition 14. Let S = {v 1 , v 1 , . . , v n , } be a finite set of n vector fields defined on a smooth manifold X. We call integral submanifold of S a submanifold Y ⊂ X whose tangent space Tp Y is spanned by the system S at every point p ∈ N . The system at every point S is integrable if through every point p ∈ X there passes an integral submanifold. Definition 15. A finite system of vector fields S = {v 1 , v 2 , . . , if ∀p(x) ∈ X, ∀i, j = 1, .
Let ω p−1 be a continuous differentiable (p−1)-form on M (Sect. 6). ,ip−1 (x), x ∈ M . 7) ∂B where d is the exterior derivative acting on forms (Definition 20). We do not provide here the algebraic details (it can be found in Sect. 6) mainly because we are interested here to underline rather the geometric interpretation of the Stokes theorem, as a representation. In that, let us remember that we can triangulate B and ∂B (Sect. 2), and obtain the sequence of chain groups Cp (B), p = 0, . . 8). ∂p−1 ∂p ∂p+1 ∂p+2 .
