uses differential forms, which require multilinear algebra and rely heavily on smooth structures. The full treatment is usually not seen until graduate school.
It is now possible to understand calculus more directly by using a theory of domains analogous to Lebesgue's theory of functions. We view domains as limits of "step chains" taken with respect to geometrically defined norms. This is analogous to viewing functions as limits of step functions, the limits taken with respect to the norm. Just as one finds more "bizarre" functions in the completed space of step functions, we find strange, new domains in the completed space of step chains. Differential forms are nothing more than linear functionals on step chains. If the linear functional is bounded with respect to the dual norm, we say the differential form is smooth. This new definition of differential form is equivalent to the classical definition using multi-vectors and leads to a theory of integration that does not use vectors, parametrizations, partitions of unity, tangent spaces, or partial derivatives, and so is not restricted to smooth domains. It is a theory of "vector calculus in without vectors" and goes beyond classical theory so that some sets with no local Euclidean structure can be treated as domains of integration.
In this talk I will sketch a proof to the special case of Stokes' theorem for surfaces with boundary that are graphs, using these ideas. I will also give examples of new domains of integration.
