Chapter XVIII: Proof of a general theorem of the infinitesimal calculus
Department of Mathematics
University of Wisconsin-Madison
dmsw@math.wisc.edu.
Copyright 2011-2017 by Philip J. Elmaa.
This book is dedicated to my mother who died in April 2016 and my father who
died in August 2012.
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Chapter XVIII: Proof of a general theorem of the infinitesimal calculus
In the previous chapter we laid down the foundation of a differential geometry on a manifold $M$ and developed the fundamental concepts
and definitions of this theory. The concepts are geometric in nature, they have to do with the description of manifolds and manifold related
manifolds. That's why I called the previous chapter a foundation of the theory.
On the other hand, the definition of vector field and flow, which are essential in the theory, are topological in nature. They
have to do with the topological properties of manifolds. In this chapter we will consider the topological nature of the fundamental
topological concept in the foundation of the theory, which is called manifold.
The notion of manifold, which is a generalization of the Euclidean manifold, was defined by Cauchy in 1829. He showed
that the Euclidean space is a Riemann manifold and that manifold is a main concept of the foundation of the theory.
However, to explain the notion of manifold we should also consider the concept of subspace. Then we will introduce the
notion of vector space, which will play a key role in defining manifold. Then we will define the local coordinates of manifold.
At the end of this chapter, we will discuss the local diffeomorphism theorem. This theorem connects the
fundamental idea of the infinitesimal calculus with the local coordinates of manifold.
The foundation of the theory of manifold will be an important problem. In the next chapter, we will begin to
study manifold and manifolds. In particular, we will review the concept of tangent space and tangent bundle.
We will also discuss the local coordinates of manifolds and manifolds. Finally, we will study the fundamental
idea of the infinitesimal calculus. That's what we mean by foundation of the theory.
Throughout this book, we will use the following notations and terminologies.
The notations we will use throughout this book are as be359ba680
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