Introduction to Analytical Geometry 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:
Cartesian plane
segments, distances, and lines in the Cartesian plane
parabolas, circumferences, ellipses, hyperbolas in the Cartesian plane
proper and improper bundles in the Cartesian plane
In addition, the main applications of these topics are mentioned and some exercises are carried out.

Leseprobe

Table of Contents

“Introduction to Analytical Geometry"

INTRODUCTION

THE CARTESIAN PLANE

THE LINE IN THE CARTESIAN PLANE

THE PARABLE IN THE CARTESIAN PLANE

THE OTHER CONICS IN THE CARTESIAN PLANE

“Introduction to Analytical Geometry"

SIMONE MALACRIDA

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

Cartesian plane

segments, distances, and lines in the Cartesian plane

parabolas, circumferences, ellipses, hyperbolas in the Cartesian plane

proper and improper bundles in the Cartesian plane

In addition, the main applications of these topics are mentioned and some exercises are carried out.

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 – THE CARTESIAN PLANE

Definitions

Translation and distance

Practical applications

Exercises

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II – THE LINE IN THE CARTESIAN PLANE

Definitions

Property

Exercises

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III – THE PARABLE IN THE CARTESIAN PLANE

Definitions

Property

Exercises

IV – THE OTHER CONICS IN THE CARTESIAN PLANE

Circumference

Ellipse

Hyperbole

General considerations on conics

Exercises

INTRODUCTION

This book presents the main results of analytic geometry, intended as an application of Euclidean geometry set in the Cartesian plane.

The conceptual evolution of analytic geometry, with respect to normal geometry, is such as to be able to begin a path which, starting from the study of polynomial functions, leads to the graphic resolution of transcendental functions (such as logarithmic, exponential, hyperbolic and trigonometric functions) up to fundamental result of mathematical analysis, i.e. the generalized study of functions of real variable.

Furthermore, analytical geometry is of fundamental contribution to the resolution of physical problems of all sorts, from those related to kinematics and mechanics to those relating to electromagnetism.

The formalism introduced by Descartes is one of the cornerstones of modern science, in all its forms.

Each chapter 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 coincides, broadly, with what is taught in technical institutes and high schools, generally in the third year.

I

THE CARTESIAN PLANE

Definitions

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Analytic geometry relates the concepts of elementary geometry with the definition of various functions expressible by means of analytic equations.

The functions can be explicit , i.e. take the form y=f(x) or implicit, in the form f(x,y)=0.

The main purpose of analytical geometry is to trace the graph of each type of function to allow a graphical display and to graphically solve the equations, a very powerful mathematical means much more than the simple formal resolution.

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The first approach to analytic geometry assumes Euclidean geometry and the definition of a Euclidean plane for plane geometry and a Euclidean space for solid geometry.

In this manual we will deal only with analytic geometry in the Euclidean plane.

The reference system in this plane is called Cartesian and consists of two oriented straight lines, perpendicular to each other, which are called Cartesian axes .

By convention, the horizontal axis is called the x-axis or x-axis, while the vertical axis is called the y-axis.