Characteristic forms in geometric integration theory

This paper develops the theory of differential forms introduced by E. Cartan. We consider the integral of a completely skew symmetric contravariant tensor T ab. . . s of rank p over a q -dimensional differentiable manifold M in a Riemannian space R n of n dimensions ( n = p + q ). We prove that the integral can be expressed in the form ∫ T ab. . . s χ ab. . . sʼ , where χ ab. . . s = ∂( α 1 , α 2 , . . . , α p )/∂( x a , x b , . . . , x s ) (─1) p , and α 1 , α 2 , . . . , α p are the characteristic functions of p ( n -dimensional) domains A 1 , A 2 , . . . , A p , whose boundaries intersect in M . The integral is taken over the whole of the space R n . The tensor χ ab . . . s is the ‘characteristic tensor’ of M . In the representation by a Grassmann algebra χ ab. . . s is expressed by a ‘characteristic form’ χ = (— 1) p d α 1 ⋀d α 2 ⋀. . . ⋀d α p , and the dual of T ab. . . s by a form U of rank q . If d denotes the exterior derivative, γ the characteristic form of C and ω the characteristic form of Ω ≡ ∂ C , then ω = -d γ . Stokes’s theorem is proved in the form — ∫d γ ⋀ U = ∫ γ ⋀d U . In the case of three-dimensional Euclidean space very simple proofs can be given by these results which form the basic theorems of ‘continuous vector analysis’ as introduced by Weyl (1940) and H. Cartan (1949).