Exercises of Tensors 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:
transformation of tensors
index rises and falls
calculation of Christoffel symbols.
Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Leseprobe

Table of Contents

"Exercises of Tensors"

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

"Exercises of Tensors"

SIMONE MALACRIDA

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

transformation of tensors

index rises and falls

calculation of Christoffel symbols.

Initial theoretical hints are also presented to make the conduct 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

Definitions

Operations

Particular tensors

Application: general theory of relativity

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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

INTRODUCTION

In this workbook, some examples of tensor calculations are carried out.

Furthermore, the main uses of these mathematical entities are presented in order to solve physical and technological problems.

In fact, tensors play a decisive role in the physical understanding of mechanics and the intrinsic properties of matter (optics and electromagnetism), even if their most famous use remains that present in the theory of general relativity and in cosmological and astrophysical applications.

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, almost exclusively in the faculties of Physics and Mathematics.

I

THEORETICAL OUTLINE

Definitions

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Given a vector space V of dimension n over a field K, the dual space of V is the vector space formed by all linear functionals mapping V to K and has dimension n.

The elements of V are called vectors, those of the dual space covectors.

We define a tensor as a multilinear map which associates h vectors and k covectors with a scalar over the field K.

Multilinearity ensures that the function is linear in each component.

A tensor thus defined has order given by the pair (h,k).

The set of all tensors of the same order gives rise to a vector space of dimension a

A tensor of order (h,k) is described by an associated matrix, called a grid, of dimension h+k.

To describe the tensor in these coordinates it is necessary to fix a basis, since different bases form different grids, therefore different components of the tensor.

Having defined a basis of V which induces a dual basis in the dual space, the following relation holds for each element of the basis:

A tensor of order (h,k) can be defined as follows in coordinates of the basis:

A tensor is independent of the choice of the basis and this will be clearly seen by introducing the product between tensors.

Given two different bases, they are connected by a base change matrix and its inverse matrix such that each element of a basis is given by multiplying the corresponding element of the change matrix (or of the inverse one) by the corresponding element of the other base.

Two tensors can be expressed completely equivalently in one basis or the other.

Given A the change of basis matrix and C the inverse matrix we have these equivalent expressions: