The idea of a differentiable manifold is a combination of ideas from both analysis and geometry.
In geometry, differentiable manifolds include such things as curves and surfaces and their higher dimensional analogues. Many of the ideas to be introduced in this chapter are in fact motivated by the study of the earth's surface in elementary geography.
In analysis, the role of differentiable manifolds is to provide a natural setting, generalizing that of normed vector spaces, in which to study differentiable functions. Thus the theory developed in Chapter 1, permits you to differentiate functions mapping one normed vector space (or some open subset thereof) into another normed vector space. But once you have studied manifolds, you will also be able to differentiate functions which map, say, a sphere into a torus.
CHARTS AND ATLASES
Although the surface of the earth is a sphere, small enough regions on it will appear flat, like a plane. A geographical atlas is in fact just a collection of maps or pictures, each of which lies in a plane.
Each such picture determines a function φ from some region U of the earth's surface into the plane R2 as shown in Figure 2.1.1. This idea leads to our first definition, in which we replace the earth's surface by an arbitrary set M and ignore nearly everything about the function φ except that it maps one-to-one onto some “flat” region — that is, an open set in some Euclidean space Rn. The integer n will be assumed fixed for the rest of this section.