Exercises of Advanced Analytical Geometry E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2023

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics: generalization of analytic geometry in the plane analytic geometry in space length and regularity of a curve parametric characterization at the geometric level Initial theoretical hints are also presented to make the performance of the exercises understood

Leseprobe

Simone Malacrida

Exercises of Advanced Analytical Geometry

BookRix GmbH & Co. KG81371 Munich

Table of Contents

Table of Contents

“Exercises of Advanced Analytical Geometry”

INTRODUCTION

GENERALIZATION OF ANALYTICAL GEOMETRY IN THE PLANE

ANALYTICAL GEOMETRY IN SPACE

LENGTH OF A CURVE

“Exercises of Advanced Analytical Geometry”

“Exercises of Advanced Analytical Geometry”

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

generalization of analytic geometry in the plane

analytic geometry in space

length and regularity of a curve

parametric characterization at the geometric level

Initial theoretical hints are also presented to make the performance of the exercises understood

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 – GENERALIZATION OF ANALYTICAL GEOMETRY IN THE PLANE

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

Exercise 1 7

Exercise 18

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II – ANALYTICAL GEOMETRY IN SPACE

Exercise 1 _

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

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III - LENGTH OF A CURVE

Exercise 1

Exercise 2

Exercise3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10 _

INTRODUCTION

INTRODUCTION

In this workbook some examples of calculations related to advanced analytical geometry are carried out, i.e. the generalization of the analytical geometry of the plane and its subsequent extension to a three-dimensional level.

Furthermore, the concepts of regularity and length of a curve are presented through the introduction of integrals and the curvilinear abscissa.

In this way it is possible to extract elementary analytic geometry by bringing the study of geometry back into mathematical analysis.

This is the first step in the construction of a differential geometry, the applications of which are innumerable from a physical point of view.

In order to understand in more detail what is explained in the resolution of the exercises, the theoretical context of reference is recalled at the beginning of each chapter.

What is stated in this workbook is generally addressed both in university geometry courses and in mathematical analysis courses (partly in analysis 1, partly in analysis 2).

I

GENERALIZATION OF ANALYTICAL GEOMETRY IN THE PLANE

GENERALIZATION OF ANALYTICAL GEOMETRY IN THE PLANE

In this chapter we will generalize what is generally presented during the study of the elementary analytic geometry of the plane.

We define versor as a vector of unit modulus , used to indicate a particular direction and verse.

Unit vectors are associated to the Cartesian axes (two in the case of the plane and three in the case of space, in this case we speak of an intrinsic triad).

In Cartesian coordinates, i.e. considering the normal Cartesian axes orthogonal to each other, the three unit vectors have the subscripts x,y,z or are called i,j,k.

Furthermore, each of them can be associated with a column vector of three components formed by 1 in the direction of the unit vector and by two 0 in the other directions.

We define direct cosines of a straight line as the cosines of the convex angles that the straight line forms with the Cartesian axes, both in the plane and in space .

If the line is oriented, the directing cosines are uniquely identified, otherwise they change sign if the orientation of the line changes.

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The general equation of a straight line in the plane considering the direction cosines is given by:

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