Introduction to Mathematical Analysis E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

The theoretical assumptions of the following mathematical topics are presented in this book:
introduction to topology
limits and calculus of limits
continuity and continuous functions
derivatives and differential calculus
integrals and integral calculus
study of functions of real variables
Each topic is treated by emphasizing practical applications and solving some significant exercises.

Leseprobe

Table of Contents

“Introduction to Mathematical Analysis"

INTRODUCTION

OUTLINE OF GENERAL TOPOLOGY

LIMITS

CONTINUITY'

DERIVATIVES AND DIFFERENTIAL CALCULUS

INTEGRALS AND INTEGRAL CALCULUS

STUDY OF FUNCTIONS WITH REAL VARIABLES

“Introduction to Mathematical Analysis"

SIMONE MALACRIDA

The theoretical assumptions of the following mathematical topics are presented in this book:

introduction to topology

limits and calculus of limits

continuity and continuous functions

derivatives and differential calculus

integrals and integral calculus

study of functions of real variables

Each topic is treated by emphasizing practical applications and solving some significant exercises.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – OUTLINE OF GENERAL TOPOLOGY

Definitions

Property

Metric, normed and Euclidean spaces

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II – LIMITS

Introduction

Definition of limit

Properties of limits

Limit theorems

Calculation of limits and notable limits

Applications

Exercises

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III - CONTINUITY

Definitions

Properties and theorems

Points of discontinuity

Exercises

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IV – DERIVATIVES AND DIFFERENTIAL CALCULUS

Definition

Property

Differential calculus

Differential calculus theorems

Geometric implications

Applications

Exercises

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V – INTEGRALS AND INTEGRAL CALCULUS

Definition

Properties and theorems

Geometric applications

Integral function and theorems

Indefinite integrals and remarkable integrals

Methods of integration

Improper integrals

Exercises

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VI – STUDY OF REAL VARIABLE VARIABLE FUNCTIONS

Scheme for the study of functions

Study of integral functions

Exercises

INTRODUCTION

In this book the main tenets of mathematical analysis are exposed.

The conceptual leap of this new sector of mathematics has been evident since its introduction, managing to extend the results previously found and probing the natural phenomena in its constitutive equations.

Mathematical analysis is in fact the fundamental prerequisite for understanding all sciences in a modern key, i.e. after the introduction of the experimental and scientific method.

Physics, chemistry, medicine, engineering, architecture, technology in general, statistics, economics and every other contemporary discipline owe mathematical analysis not only the correct setting of problems, but also the resolution of the same through equations and solutions that can be understood after the necessary concepts introduced in this manual.

The practical applications of this mathematical formalism are therefore absolutely essential with the society of the last four centuries.

Each of the chapters will be accompanied by some final exercise. This manual is not a workbook and, precisely for this reason, you will not find hundreds of exercises.

The questions proposed were considered significant for understanding the main rules and for their application.

In addition, particular emphasis has been given to the method of solving them since the real leap in quality between the study of a rule and its application is given precisely by the method, i.e. by the quality of the reasoning, and not by the quantity of calculations.

The program presented in this manual expands what was taught in the last year of scientific high schools, coinciding with almost all of the topics presented in the first university course of mathematical analysis.

I

OUTLINE OF GENERAL TOPOLOGY

Definitions

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The conceptual leap between elementary and advanced mathematics was evident only after the introduction of mathematical analysis.

The fact that this discipline was local, and not punctual, led to the study and development of topology, understood as the study of places and spaces not only in a geometric sense, but in a much broader sense.

The general topology gives the foundations of all the underlying sectors, among which we can include the algebraic topology, the differential one, the advanced one and so on.

We define topology as a collection T of subsets of a general set X for which the following three properties hold:

1) The empty set and the general set X belong to the collection T.

2) The union of an arbitrary quantity of sets belonging to T belongs to T.

3) The intersection of a finite number of sets belonging to T belongs to T.

A topological space is defined with a pair (X, T) and the sets constituting the collection T are open sets.

Particular topologies can be the trivial one in which T is formed by X and the empty set and the discrete one in which T coincides with the set of parts of X.

In the first topology only the empty set and X are open, while in the discrete one all sets are open.

Two topologies are comparable if one of them is a subset of the other, while if one topology contains the other, the first is said to be finer than the second.

The set of all topologies is partially ordered: the trivial topology is the least fine, the discrete is the finest, and all other possible topologies have intermediate fineness between these two.

In a topological space, a set I containing a point x belonging to X is called (open) neighborhood of x if there exists an open set A contained in I containing x:

A subset of a topological space is closed if its complement is open.

Closed sets have three properties:

1) The union of a finite number of closed sets is a closed set.

2) The intersection of closed sets is a closed set.

3) The set X and the empty set are closed.

With these properties, a topology based on closed sets can be constructed.

In general, a subset can be closed, open, both open and closed, neither open nor closed.

Said S a subset of a topological space X, x is a point of closure of S if every neighborhood (open or closed) of x contains at least one point of S.

Said S a subset of a topological space X, x is an accumulation point of S if every neighborhood (open or closed) of x contains at least one point of S different from x itself.

Each accumulation point is a closing point while vice versa is not valid.