Differential Calculus and Its Applications

Michael J. Field

• Éditeur: Dover Publications
• Année: 2013
• ISBN:
  9780486298849
• Sujet: Mathematik
• Sous-sujet: Infinitesimalrechnung

This text offers a synthesis of theory and application related to modern techniques of differentiation. Based on undergraduate courses in advanced calculus, the treatment covers a wide range of topics, from soft functional analysis and finite-dimensional linear algebra to differential equations on submanifolds of Euclidean space. Suitable for advanced undergraduate courses in pure and applied mathematics, it forms the basis for graduate-level courses in advanced calculus and differential manifolds.
Starting with a brief resume of prerequisites, including elementary linear algebra and point set topology, the self-contained approach examines liner algebra and normed vector spaces, differentiation and calculus on vector spaces, and the inverse- and implicit-function theorems. A final chapter is dedicated to a consolidation of the theory as stated in previous chapters, in addition to an introduction to differential manifolds and differential equations.

CHAPTER 1

Linear algebra and normed vector spaces

1.1 Normed vector spaces

In this chapter K will always denote either the field of real numbers, R, or the field of complex numbers, C. | | will denote the absolute value or modulus of numbers in K.

Let E be a vector space over K of not necessarily finite dimension. As it stands, E is too general to admit of interesting study. Thus, without any topology on E, questions like ‘Is vector space addition a continuous operation?’ or ‘Are linear maps continuous?’ are meaningless. We therefore wish to start by studying vector spaces with some additional structure such as a topology or metric. One such structure that the reader will have already encountered in the study of the vector spaces R, R², R³ and C is the important notation of the length of a vector. The generalization of length to an arbitrary vector space is given by the following definition.

Definition 1.1.1

Let E be a vector space over the field K. A function

is called a norm on E if

1.

x

= 0 if and only if x = 0

2.

for all x, y ∈ E (‘triangle inequality’)

3.

for all x ∈ E and k ∈ K.

The pair (E,

) is then called a normed vector space.

A norm, then, is our generalization of length. The following proposition shows that the norm of a vector is, like length, always positive.

Proposition 1.1.2

Let (E,

) be a normed vector space. Then

x

0 for all x ∈ E...