ABSTRACT
This chapter discusses the important notions of immersions, submersions, and submanifolds as qualifiers of how manifolds may relate to one another. A key concept in analysis is the ability to take derivatives. The chapter describes the definition of the Lie derivative to tensor fields of all types, and not just functions and tangent vector fields. It introduces the linear algebra necessary for differential forms. The chapter explains the notion of a pull-back of a covariant tensor fields by a smooth function between manifolds. It presents the theory of integration on manifolds, and calculations and applications with integration. The chapter also discusses Stokes’ Theorem. Readers may be aware of the difference between Riemannian integration, the theory introduced in the usual calculus sequence, and Lebesgue integration, which relies on measure theory. The chapter concludes with a brief comment about the algebra of differential forms.
