Exercises of Multi-Variable Functions E-Book

Simone Malacrida

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

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
real functions with several variables
search for constrained maxima and minima
remarkable theorems 
Initial theoretical hints are also presented to make the performance of the exercises understandable.

Leseprobe

Table of Contents

“Exercises of Multi-Variable Functions”

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

“Exercises of Multi-Variable Functions”

SIMONE MALACRIDA

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

real functions with several variables

search for constrained maxima and minima

remarkable theorems

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

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 – THEORETICAL OUTLINE

Introduction

Operations

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II – EXERCISES

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

INTRODUCTION

In this workbook, some examples of calculations relating to real functions with several variables are carried out.

Furthermore, the main theorems used in this sector of differential analysis and their practical use in order to solve problems are presented.

Multi-variable real functions not only represent a generalization of single-variable real functions, but coincide with a fundamental mathematical support for solving various physical and applicative problems.

The introduction of concepts such as differentiability and operations relating to the algebra of nabla, as well as the contribution to the solution of constraint reactions, make this chapter of the analysis one of the most fruitful within the panorama of mathematics.

In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.

What is presented in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2).

I

THEORETICAL OUTLINE

Introduction

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Functions of real variables with several variables are an extension of what has been said for real functions with one variable.

Almost all the properties mentioned for one-variable functions remain valid (such as injectivity, surjectivity and bijectivity), except the ordering property which is not definable.

The domain of a multivariate function is given by the Cartesian product of the domains calculated on the single variables.

A level set , or level curve, is the set of points such that:

The level set with c=0 is used to analyze the sign of the function in the domain.

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Operations

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The topological definition of limit is the same as that given for one-variable functions, the metric definition changes as follows:

The limit exists if its value does not depend on the direction in which it is calculated.

The same applies to continuity.

A function is said to be continuous separately with respect to one of its variables if it is continuous as a function of the single variable, keeping the other constants.

Separate continuity is a weaker condition than global continuity across all variables.

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For a function of several variables, however, there are different concepts of derivative.

We call partial derivative the derivative carried out only on one of the variables, always defining the derivative as the limit of an incremental ratio.

To distinguish the partial derivative from the total one, the symbol is used .

Partial derivatives of higher order return the order to the exponent of that symbol.

A point is said to be simple if the first partial derivatives are continuous and not zero, but if one of the derivatives is zero or does not exist, the point is said to be singular.

Partial differentiability implies separate continuity.

By extending the concept of partial derivative from a path along the coordinate axes to any path, we have the directional derivative.

Once a generic unit vector is defined, the directional derivative along that vector is given by: