Stokes' theorems in advanced calculus

Abstract

We explicitly show that Green's, Gauss's, and Stokes' theorems in advanced calculus are indeed corollaries of the Stokes' theorem on manifolds. To understand Stokes' theorem on manifolds properly we survey first of all manifold theory itself which starts from the definition of smooth manifold, and develops such notions as tangent vector space, smooth vector bundle, exterior derivative, manifold with boundary, Riemannian metric, orientation, induced orientation, integration of forms etc.. We also need to interpret such notions as line integral, surface integral, divergence and curl which concern the vector field into the concepts such as integrations of forms and exterior derivatives which concern the forms. With these preparations we derive the Stokes' theorems in advanced calculus from the Stokes' theorem on manifolds.