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2023 | Buch

Sets of Numbers and Metric Spaces Kapitel lesen Erstes Kapitel lesen

A Modern Introduction to Mathematical Analysis

verfasst von: Alessandro Fonda

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook presents all the basics for the first two years of a course in mathematical analysis, from the natural numbers to Stokes-Cartan Theorem.
The main novelty which distinguishes this book is the choice of introducing the Kurzweil-Henstock integral from the very beginning. Although this approach requires a small additional effort by the student, it will be compensated by a substantial advantage in the development of the theory, and later on when learning about more advanced topics.
The text guides the reader with clarity in the discovery of the many different subjects, providing all necessary tools – no preliminaries are needed. Both students and their instructors will benefit from this book and its novel approach, turning their course in mathematical analysis into a gratifying and successful experience.

Inhaltsverzeichnis

Frontmatter

The Basics of Mathematical Analysis

Frontmatter
1. Sets of Numbers and Metric Spaces
Abstract
In this chapter, we introduce the main settings where all the theory will be developed. First, we discuss the sets of numbers \({\mathbb N}\), \({\mathbb Z}\), \({\mathbb Q}\), \({\mathbb R}\), and \({\mathbb C}\), then the space \({\mathbb R}^N\), and, finally, abstract metric spaces.
Alessandro Fonda
2. Continuity
Abstract
In this chapter we introduce one of the most important concepts in mathematical analysis: the “continuity” of a function. This topic will be treated in the general framework of metric spaces.
Alessandro Fonda
3. Limits
Abstract
We will now introduce another fundamental concept that, however, is strongly related to continuity. It is the notion of the “limit” of a function, a local notion, as we will see. As in Chap. 2, the theory will be developed within the framework of metric spaces.
Alessandro Fonda
4. Compactness and Completeness
Abstract
In this chapter we discover some more subtle properties of the set of real numbers. This investigation will emphasize two important concepts, which will then be analyzed in the general setting of metric spaces: compactness and completeness.
Alessandro Fonda
5. Exponential and Circular Functions
Abstract
The aim of this chapter is to provide a unified construction of the exponential and the trigonometric functions using geometrical arguments in the complex plane. The basis of this construction will lie in the proof of the following statement.
Alessandro Fonda

Differential and Integral Calculus in ℝ

6. The Derivative
Abstract
We start by introducing the concept of “derivative” of a function defined on a subset of \({\mathbb R}\), taking its values in \({\mathbb R}\).
Alessandro Fonda
7. The Integral
Abstract
.In this chapter, we denote by I a compact interval of the real line \({\mathbb R}\).
Alessandro Fonda

Further Developments

Frontmatter
8. Numerical Series and Series of Functions
Abstract
Let V  be a normed vector space. Given a sequence (a k)k in V , the associated “series” is the sequence (s n)n defined by
$$\displaystyle \begin {array}{lll} &&s_0=a_0\,,\\ &&s_1=a_0+a_1\,,\\ &&s_2=a_0+a_1+a_2\,,\\ &&\; \dots \\ &&s_n=a_0+a_1+a_2+\dots +a_n\,,\\ &&\; \dots \end {array} $$
Alessandro Fonda
9. More on the Integral
Abstract
Let us further analyze the definition of integral for a function \(f:I\to {\mathbb R}\) when I = [a, b] is a compact interval.
Alessandro Fonda

Differential and Integral Calculus in ℝ N

10. The Differential
Abstract
Let \(\mathcal {O}\subseteq {\mathbb R}^N\) be an open set, x 0 a point of \(\mathcal {O}\), and \(f:\mathcal {O}\to {\mathbb R}^M\) a given function.
Alessandro Fonda
11. The Integral
Abstract
In this chapter we extend the theory of the integral to functions of several variables defined on subsets of \({\mathbb R}^N\!,\) with values in \({\mathbb R}.\) For simplicity, in the exposition we will first focus our attention on the case N = 2 and later provide all the results in the case of a generic dimension N.
Alessandro Fonda
12. Differential Forms
Abstract
Let us start considering the projections in \({\mathbb R}^N\); as we have already seen, these are the functions \(p_m:{\mathbb R}^N\to {\mathbb R}\) defined by.
Alessandro Fonda
Backmatter
Metadaten
Titel
A Modern Introduction to Mathematical Analysis
verfasst von
Alessandro Fonda
Copyright-Jahr
2023
Electronic ISBN
978-3-031-23713-3
Print ISBN
978-3-031-23712-6
DOI
https://doi.org/10.1007/978-3-031-23713-3

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