6533b82efe1ef96bd129304c

RESEARCH PRODUCT

Integration on Surfaces

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We intend to study the integration of a differential k-form over a regular k-surface of class C 1 in \(\mathbb{R}^n\). To begin with, in Sect. 7.1, we undertake the integration over a portion of the surface that is contained in a coordinate neighborhood. Where possible, we will express the obtained results in terms of integration of vector fields. For example, we study the integral of a vector field on a portion of a regular surface in \(\mathbb{R}^3\) and also the integral over a portion of a hypersurface in \(\mathbb{R}^n\). In Sect. 7.3 we study the integration of differential k-forms on regular k-surfaces admitting a finite atlas.We discuss the need for the surface to be orientable so that the defined integral makes sense in this more general situation. Although the requirement of having a finite atlas seems rather restrictive, all compact surfaces fall into this category, as do almost all the surfaces that one might naturally encounter (including many that are not compact). Thus, in the context of vector calculus, where applications play a key role, this restriction is not as great as it first appears.