Calculus IV with Differential Geometry
This course extends techniques from Calculus III by studying scalar and vector fields in n-dimensional spaces and operations on them. The notions of line and surface integrals are introduced, and Green’s, Stokes’ and Gauss’s theorems and their applications are discussed. Starting with parametrized surfaces in R3, this course introduces the concepts of embedded manifolds, tangent spaces, and tangent bundles as well as Gauss curvature for two-dimensional surfaces. The notion of differential forms on manifolds is developed, and the general Stokes’ theorem for forms is formulated at the end of the course.