Exercises of Differential Forms 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:
exact differential forms and primitives
conservative fields and potentials
remarkable theorems of differential analysis
Initial theoretical hints are also presented to make the performance of the exercises understood.

Leseprobe

Table of Contents

“Exercises of Differential Forms”

INTRODUCTION

REMARKABLE THEOREMS

CONSERVATIVE AND POTENTIAL FIELDS

EXACT AND PRIMITIVE FORMS

“Exercises of Differential Forms”

SIMONE MALACRIDA

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

exact differential forms and primitives

conservative fields and potentials

remarkable theorems of differential analysis

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

––––––––

INTRODUCTION

––––––––

I – REMARKABLE THEOREMS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

––––––––

II – CONSERVATIVE AND POTENTIAL FIELDS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

––––––––

III - EXACT AND PRIMITIVE FORMS

Exercise 1

Exercise 2

Exercise 3

INTRODUCTION

In this workbook some examples of calculations related to differential, exact and primitive forms and to conservative and non-conservative vector fields are carried out.

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

Differential forms, together with conservative fields, play a first-rate role in physics and chemistry, describing particular natural entities and actively participating in the constitution of the basic equations of mechanics, electromagnetism, thermodynamics and quantum physics .

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

What is stated in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2) as a preparatory practice for physics courses.

I

REMARKABLE THEOREMS

The exercises presented in this chapter are solved using the relevant theorems relating to differential forms, in particular Green's theorem, Stokes' and Gauss's theorem.

Let us briefly recall the theoretical assumptions.

Given a curve parametrized by a smooth function and a piecewise smooth plane curve that is simple and closed, the curl theorem holds for a vector field:

This theorem relates the line integral to the surface integral.

The theorem is a generalization of the fundamental theorem of integral calculus and is related to the definitions of conservatism and irrotality of a vector field.

A special case of the rotor theorem is Green's theorem .

Given a piecewise regular closed simple curve which is the boundary of a surface and given two real functions of two real variables which have continuous partial derivatives on an open set which contains the surface then we have that:

This theorem is therefore the curl theorem in the two-dimensional case.