Analysis on Manifolds

In Chapter 3, we introduced the concept of a differentiable manifold as motivated by a search for topological spaces over which it is possible to do calculus or do physics. The idea of having a topological space locally homeomorphic to Rn drove the definition of a differentiable manifold. Subsequent sections in that chapter discussed differentiable maps between manifolds and the corresponding differential. We used these to introduce the important notions of immersions, submersions, and submanifolds as qualifiers of one way manifolds can relate to one another.