## Abstract

This paper develops the theory of differential forms introduced by E. Cartan. We consider the integral of a completely skew symmetric contravariant tensor T$^{ab\ldots s}$ 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 $\intT^{ab\ldotss}\chi_{ab\ldotss},$ where $\chi_{ab\ldotss} = \frac{\delta(\alpha_1, \alpha_2, \ldots, \alpha_p)}{\delta(x^a, x^b, \ldots , x^s)}(-1)^p,$ and $\alpha_1$, $\alpha_2$, $\ldots$, $\alpha_p$ are the characteristic functions of p (n-dimensional) domains A$_1$, A$_2$, $\ldots$, A$_p$, whose boundaries intersect in M. The integral is taken over the whole of the space R$^n$. The tensor $\chi_{ab\ldotss}$ is the `characteristic tensor' of M. In the representation by a Grassmann algebra $\chi_{ab\ldotss}$ is expressed by a `characteristic form' $\chi = (-1)^p d\alpha_1\Lambdad\alpha_2\Lambda\ldots\Lambdad\chi_p,$ and the dual of T$^{ab\ldotss}$ by a form U of rank q. If d denotes the exterior derivative, $\gamma$ the characteristic form of C and $\omega$ the characteristic form of $\Omega$ \equiv $\delta C$, then $\omega$ = -d$\gamma$. Stokes's theorem is proved in the form $-\int d\gamma\Lambda U = \int\gamma\LambdadU.$ 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).