Stokes' theorem over simplexes via the generalized Riemann integral

Abstract

The generalized Riemann integral is similar to the ordinary Riemann integral, yet it integrates a much wider class of functions and has much nicer properties. One very nice property is the fundamental theorem of calculus, which does not require the assumption that the derivative F' be integrable in order to obtain the basic formula (GR) aʃb F'(x) dx = F(b) - F(a) ; instead, the integrability of F' is one of the conclusions of the theorem. In this research, we give a simple, concrete definition of an integral of a function over a k-simplex in Rk such that (i) we are able to prove a version of Stokes' theorem for arbitrary differential forms which are differentiable, and (ii) if k = 1, then the integral is the same as the generalized Riemann integral over a closed interval in R. The main result of this thesis is the following version of Stokes' theorem : Let k, n {u1D716} Z+ be such that k ≤ n. Let Ω be a nonempty open subset of Rn , and let Ω be a nonempty open subset of Rn, and let {u1D6D4} = [P0,P1, ….,Pk] be an oriented affine k-simplex in Ω. If is a differentiable (k-1)-form on Ω, then ʃσ d{u1D714} exists and ʃσ d{u1D714} = ʃᴂ{u1D714}