Mathematical Methods 4 Electrotechnic Freaks E-Book

Jürgen Ulm

• Herausgeber: expert verlag GmbH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 298

Beschreibung

The book offers a practice-oriented introduction to the mathematical methods of electrical engineering. The focus is on the solution of ordinary and partial differential equations using analytical and numerical methods. The analytical methods are opposed to the numerical methods. The differential equations were chosen with a view to the problems of electrical engineering. It is shown how they can also be transferred to mechanics or thermodynamics. Numerous examples and exercises with elaborated solutions facilitate the transfer of knowledge to applications.

Leseprobe

Jürgen Ulm

Mathematical Methods 4 Electrotechnic Freaks

Umschlagabbildung: © Jürgen Ulm

Bibliografische Information der Deutschen Nationalbibliothek

Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar.

DOI: https://doi.org/10.24053/9783381116522

© 2023 · expert verlag

‒ ein Unternehmen der Narr Francke Attempto Verlag GmbH + Co. KG

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ISBN 978-3-381-11651-5 (Print)

ISBN 978-3-381-11652-2 (ePDF)

ISBN 978-3-381-11653-9 (ePub)

Foreword

Mathematics is the universal tool for the scientist,

„… for the mathematic is the basis of all exact scientific knowledge…“

(David Hilbert, German mathematician, 1862-1943).

Special attention is therefore paid to learning how to use the tool. As is so often the case, the realisation of the necessity paired with the motivation of the user is in the foreground. If the declared aim is to describe physical relationships by means of mathematics, this does not necessarily require thematic rigour. The application of mathematical rigour is likely to be counterproductive to this concern. Furthermore, Gödel’s incompleteness theorem of mathematics applies, which even shows mathematics itself its limitations. Experience has shown that the users’ desire for mathematical rigour can be observed when they are convinced and enthusiastic about mathematics and its possibilities. For this reason, mathematical rigour should not be given the highest priority at the beginning. Mathematics lives from the joy of its users and applications!

„It is impossible to adequately convey the beauties of the laws of nature if someone does not understand mathematics. I regret that, but it is probably so.“

(Richard Feynman, physicist and Nobel Prize winner, 1918 1988),

denn

„The book of nature is written in the language of mathematics.“

(Galileo Galilei, 1564 1642).

Calculator, paper, pencil and eraser in combination with coffee form a good basis. Mathematics is the universal tool of electrical engineering. Selected mathematical methods are also used to deal with selected topics in electrical engineering. The work is carried out by presenting the basics, describing the task and solving the problem in detail. The target group of readers also results from this procedure. From the author’s point of view, these are:

•Students of engineering sciences who would like to work on scientific topics using mathematical methods.

•Software engineers who want to implement differential equations in matrix form in microprocessors.

•Simulation engineers who would like to calculate something „on foot“.

•Measurement engineers who need a measurement value from a location where no sensor can be adapted and only calculations can be made for this location.

•Maths brave, pale in the face, survived and now want to try maths again.

Since our science has a mirror-image structure, it is worthwhile, for example, to familiarise oneself in depth with a scientific discipline. Here, electrical engineering is preferably recommended. By changing the coefficients of a differential equation, the enthusiastic reader of this book conquers another scientific discipline (hence the use of the term “mirror image”). For example, anyone who can solve electrical networks (meshes) can consequently also solve thermal, magnetic, mechanical and hydraulic networks. The mathematical basics include calculation rules, definitions, matrices, ordinary and partial differential equations and coordinate systems. They provide access to understanding the chosen mathematical methods and applications in electrical engineering. An elementary application in electrical engineering is the LCR oscillating circuit, which is described with differential equations and whose properties are presented. The integral transformation, the method of moments and Green’s method have in common the formation of the inner product for the solution of differential equations. The last two methods are introduced in detail with the help of examples. With the method of moments, the transition to the finite element method (FEM) and finite difference method (FDM) is made using application examples. The method of moments is also used to introduce the eigenvalue problem. The development of infinite series by alternately applying the law of flow and the law of induction leads to Bessel functions as well as to the phenomenon of field displacement with the effect of current displacement in the conductor. Selected standards should provide the reader with hints for the preparation of scientific documentation. A note on the extended use of the book is permitted: New exercises can be generated by simply modifying the original problem that has already been solved. The modification of the original task should be done in such a way that its solution is already known in advance. This gives the possibility to compare the results and to further deepen the familiarisation. Because the following always applies

„Uncertain are the calculations of the dispersible“

(Wisdom Literature).

With kind regards the

author

autumn 2023

For more information on the institutes, see also Appendix B.

Contents

1Required mathematical basics

1.1Logarithm

1.2Matrices

1.2.1Arithmetic operations with matrices

1.2.2Addition and subtraction of two matrices

1.2.3Multiplication of a matrix with a scalar

1.2.4Square matrix

1.2.5Identity matrix

1.2.6Determinant

1.2.7Subdeterminant or minor

1.2.8Adjuncts or algebraic complement

1.2.9Inverse matrix

1.2.10Transposed of a matrix

1.2.11Complex conjugate matrix

1.2.12Hermite conjugate matrix

1.2.13Hermitian matrix – self-adjoint matrix

1.2.14Orthogonal matrix

1.2.15Unitary matrix

1.2.16Normal matrix

1.2.17Norm of a matrix

1.2.18Conditioned matrix equation and condition number

1.2.19Eigenvalue, eigenvector

1.2.20Square matrices – a summary

1.3Integral, differential equations

1.3.1Definitions

1.3.2Differentiation of scalar functions

1.3.3Higher order ordinary differential equations

1.3.4Partial differential equations

1.3.5Partial integration

1.3.6Classification of differential equations

1.3.7Initial value task

1.3.8Boundary value problem

1.3.9Linear operators

1.3.10Inner product

1.3.11Strong form/formulation of a differential equation

1.3.12Weak form/formulation of a differential equation

1.4Vector classification

1.5Differentiation rules for vectors

1.6Vector operators

1.6.1Nabla and Laplace operator

1.6.2Vector operator Gradient

1.6.3Vector operator Divergence

1.6.4Vector operator Curl

1.6.5Comparison of vector operators

1.6.6Rules of calculation for the Nabla operator

1.6.7Comparison scalar and vector product

1.6.8Base, unit vectors

1.7Boundary operator ∂

1.8Maxwell’s equations

1.8.1Relationship between circular and surface integral

1.8.2Relation between area integral and volume integral

1.8.3Maxwell’s equations – differential form

1.8.4Maxwell’s equations – integral form

1.8.5Directional assignment of involved vector fields

1.9Dirac’s delta function

2Coordinate systems

2.1Cartesian coordinate system

2.2Cylinder coordinate system

2.3Sphere coordinate system

3Geometric mean distance – GMD

3.1Geometric mean distance – what for?

3.2Geometric mean distance – definitions and basics

3.2.1Euclid – The Elements (extracts)

3.2.2Arithmetic means – definition

3.2.3Geometric mean – definition

3.2.4GMD – possible combinations

3.2.5GMD – graphical interpretation

3.2.6Why geometric mean?

3.3GMD of two collinear lines

3.3.1GMD calculation – numerical solution

3.3.2GMD calculation – analytical solution

3.3.3GMD calculation – example

3.4GMD of a collinear arrangement between a point and a line

3.4.1GMD calculation – numerical solution

3.4.2GMD calculation – analytical solution

3.4.3GMD calculation – example

3.5GMD of a line on itself

3.5.1GMD calculation – analytical solution

3.5.2GMD calculation – numerical solution

3.5.3GMD calculation – summary

3.6GMD of two parallel lines

3.6.1GMD calculation – numerical solution

3.6.2GMD calculation – analytical solution

3.6.3GMD calculation – example

3.7GMD of a point and a helix

3.7.1Length of an unwound helix

3.7.2GMD calculation – analytical solution

3.8GMD point outside line with its perpendicular on line centre

3.8.1GMD calculation – numerical solution I

3.8.2GMD calculation – numerical solution II

3.8.3Analytical solution and example calculation

3.8.4GMD calculation – summary

3.9GMD point outside line with its perpendicular on line end

3.9.1GMD calculation – radius right at the element

3.9.2GMD calculation – radius left at the element

3.9.3GMD calculation – analytical solution

3.9.4GMD calculation – summary and evaluation

3.10GMD point outside line with its perpendicular inside line

3.10.1GMD calculation – radius right at the element

3.10.2GMD calculation – radius left at the element

3.10.3GMD calculation – superposition

3.10.4GMD calculation – analytical solution

3.10.5GMD calculation – Summary and evaluation

4LCR parallel and series resonant circuit

4.1Resonant circuits, impedances and resonances

4.2Natural frequency – error calculation

4.3Voltage profiles LCR series resonant circuit with frequency variation

4.3.1Voltage characteristics across the inductance

4.3.2Voltage characteristics across inductance and resistance

4.3.3Voltage characteristics across the resistor

4.3.4Voltage characteristics across capacitance

4.4Damped forced LCR series resonant circuit

4.5Damped free LCR series resonant circuit

4.6Undamped free LC resonant circuit

4.7Damped forced LCR parallel resonant circuit

4.8Damped free LCR parallel resonant circuit

4.9Undamped free LC resonant circuit

5Current displacement in conductor

5.1Current displacement in the conductor – modelling

5.2Current displacement in the conductor – calculation result

5.3Current displacement in the conductor – simulation result

5.4Current displacement in conductors – summary

6Bessel equation and Bessel function

6.1On the person Wilhelm Friedrich Bessel

6.2Bessel equation and solutions

6.3Bessel equation of the field diffusion equation

6.4Bessel function for calculating the field distribution in a capacitor

6.4.1Model arrangement

6.4.2Derivation of the Bessel function

6.5Bessel function for calculating the flux density within a coil

6.5.1Model arrangement

6.5.2Derivation of the Bessel function

6.6Bessel function from general form of Bessel equation

7Solution of differential equations using Green’s functions

7.1About George Green

7.2Green’s integral theorems

7.3PDE – arrangements of evaluation points and integration points

7.4PDE – preparation for solution by Green’s – differential form

7.5PDE – preparation for solution by Green’s – integral form

7.5.1Converting the PDE according to the variable to be solved

7.5.2Homogeneous boundary conditions

7.5.3Inhomogeneous boundary conditions

7.5.4Dirichlet boundary conditions

7.5.5Neumann boundary conditions

7.6PDE – solution of Poisson’s DGL

7.6.1Exercise description

7.6.2Solution path

7.7PDE – solution of Laplace’s DGL

7.7.1Exercise description

7.7.2Solution path

7.8ODE – Preparation for the solution with the Green’s function

7.8.1Homogeneous boundary conditions

7.8.2Inhomogeneous boundary conditions

7.8.3Continuity and discontinuity conditions

7.9.1Exercise description

7.9.2Solution I

7.9.3Solution II

7.10.1Exercise description

7.10.2Solution

7.11.1Exercise description

7.11.2Solution path

7.12.1Exercise description

7.12.2Solution path

7.13.1Exercise description

7.13.2Solution path

8Method of Lagrangian multipliers

8.1Definition of the Lagrange multiplier method

8.1.1Properties of the method

8.1.2Mathematical optimisation

8.1.3Calculus of variations

8.2Derivation of the Lagrange multiplier method

8.3Application of the method

8.4Maths example – extreme value problem with one constraint

8.5Maths example – extreme value problem with two constraints

8.6Application example – cube inscribed in a sphere

8.6.1Extreme value problem with one constraint

8.6.2Solution with Lagrange multiplier method

8.6.3Solution with elimination method

8.7Application example – dimensioning of a coil winding

8.7.1Extreme value problem

8.7.2Solution procedure

9Differential equations and finite elements

9.1Physics examples for differential equations of 1′th order

9.2Physics examples for 2′th order differential equations

9.3Finite elements

10From the Method of Moments to the Galerkin Method

10.1Basic principle of the method of moments (MOM)

10.2Remarks on the method of moments

10.2.1Matrix (ljk)

10.2.2Choosing the basis and weighting functions ϕn and wk

10.3About Boris Galerkin

10.4Galerkin’s idea

11Traditional Galerkin Method

12.1Choosing the base and weighting function

12.2Weak formulation of the differential equation

12.3Transforming the system of equations into a matrix equation

12.4Solving the linear equation system

13.1Choosing the base and weighting function

13.2Formulation of the weak form with basis and weighting function

13.3Transforming the system of equations into a matrix equation

13.4Solving the linear equation system

14.1Choosing the base and weighting function

14.2Weak formulation of the differential equation

14.3Transforming the system of equations into a matrix equation

14.4Solving the linear equation system

15.1Choosing the base and weighting function

15.2Weak formulation of the differential equation

15.3Transforming the system of equations into a matrix equation

15.4Solving the linear equation system

16Galerkin method – Ampere’s law

16.1Galerkin method – Ampere’s law for the conductor inside

16.1.1Weak formulation of the differential equation

16.1.2Transforming the system of equations into a matrix equation

16.1.3Solving the linear equation system

16.2Galerkin method – Ampere’s law for the conductor outside

16.2.1Weak formulation of the differential equation

16.2.2Transforming the system of equations into a matrix equation

16.2.3Solving the linear equation system

16.3Comparison of FEM with Galerkin results

17Galerkin-FEM

17.1Galerkin FEM – What is being solved?

17.2Galerkin-FEM – Procedure for the solution

18.1Weak formulation of the differential equation

18.2Discretisation of the domain Ω to be solved

18.3Choosing the base and weighting function

18.4Formulation of the weak form with triangular functions ϕ(x)

18.5Transforming the system of equations into a matrix equation

18.6Solving the linear equation system

19.1Weak formulation of the differential equation

19.2Discretisation of the domain Ω to be solved

19.3Choosing the base and weighting function

19.4Formulation of the weak form with triangular functions ϕ(x)

19.5Transforming the system of equations into a matrix equation

19.6Solving the linear equation system

20Galerkin-FEM – Electrostatic field calculation

20.1Weak formulation of the differential equation

20.2Discretisation of the domain Ω to be solved

20.3Choosing the base and weighting function

20.4Formulation of the weak form with triangular functions ϕ(x)

20.5Transforming the system of equations into a matrix equation

20.6Solving the linear equation system

21Galerkin-FEM – heat diffusion

21.1Weak formulation of the differential equation

21.2Discretisation of the domain Ω to be solved

21.3Choosing the base and weighting function

21.4Formulation of the weak form with triangular functions ϕ(x)

21.5Transforming the system of equations into a matrix equation

21.6Solving the linear equation system

21.7Diffusion process completed

22Galerkin-FEM – magnetic field diffusion

22.1Weak formulation of the differential equation

22.2Discretisation of the domain Ω to be solved

22.3Choosing the base and weighting function

22.4Formulation of the weak form with triangular functions ϕ(x)

22.5Transforming the system of equations into a matrix equation

22.6Solving the linear equation system

23Introduction to the finite difference method

23.1Numerical notation of the linear field diffusion equation

23.2On the persons Crank and Nicolson

23.3Solution with implicit method according to Crank-Nicolson

23.3.1Transforming the diffusion equation into a matrix equation

23.3.2Solving the matrix equation

23.3.3Application example

23.4Solution with explicit method according to Crank-Nicolson

23.4.1Transforming the diffusion equation into a matrix equation

23.4.2Solving the matrix equation

23.4.3Application example

24Applications of FEM to product development

24.1Analysis of a proportional magnet

24.1.1Preprocessing

24.1.2Processing

24.1.3Postprocessing

24.2Synthesis of a planar asynchronous disc motor

24.2.1Preprocessing

24.2.2Processing

24.2.3Postprocessing

24.2.4Prototype of the planar asynchronous motor

25Virtual product design

25.1Coupling between FEM and optimisation tools

25.2Multi-objective optimisation – Pareto optimisation

25.3Optimisation example electromagnet

25.3.1Monte Carlo method

25.3.2Particle swarm method

25.3.3Evolutionary method

25.3.4Discussion of the results

26Eigenvalue problems

26.1Eigenvalue problem – introduction

26.2Eigenvalue problem – method of moments

26.3Eigenvalue problem – canonical form

27.1Exercise description

27.2Solution path and solution

27.3Solution for 1′th order

27.4Solution for 2′th order

28Common features of methods to solve differential equations

28.1Method of Moments (MOM)

28.2Integral transformation

28.3Green’s method

29Things worth knowing about modelling

29.1Categories of modelling

29.2Analytics versus Numerics

30Useful standards

Bibliography

AAppendix

A.1Integrals

A.2Integrals for chap. 3.3

A.3Integrals for chap. 3.5

A.4Integrals for chap. 3.6

A.5MATLAB-Code – Heat diffusion script

A.6MATLAB code – magnetic field diffusion script

A.7Tool comparison – MATLAB vs. COMSOL

BCampus Künzelsau – Inside

Index

Symbols and abbreviations

Symbol

Meaning

Unit

A

coefficient, matrix

A

area

m2

B

coefficient, matrix

B,

magnet. flux density, vector of magnet. flux density

V s/m2

Bh

interpolation, approach function

C

coefficient, matrix

C

capacity

As/V

C

heat capacity

J/K

D

coefficient, Charge

D

charge density

As/m2

D

discriminant

E

coefficient, matrix

E,

electric field strength, electric field strength

V/m

ε

length-related electric field strength

V/m2

F

coefficient, function

F

force

N, kgm/s2

G

Green’s function

G

coefficient

H,

magnet. field strength, vector of the magnet

A/m

HΦ

field interpolation function, approach function

I

current

A

J,

electr. current density, vector of electr. current density

A/m2

K

constant

L

inductivity

Vs/A

M

matrix

N

number of nodes, line elements, running variable, number of turns

P

power

W

P

polynomial function, evaluation point

P′

source point, integration point

Q

charge

As

R

residuum

R

radius

m

R

resistance

Ω

S

matrix

SP

vertex

U

voltage

V

V

volume

m3

W

Wronski determinant

X

reactance, reactance

Ω

Z, | Z |

impedance, magnitude of the impedance

Ω

Z

impedance (complex impedance)

Ω

a

coefficient

a0

acceleration

m/s2

b

damping coefficient

kg/s

c

constant

c

spring constant

N/m

c

speed of light

m/s

c

specific heat capacity

J/(kgK)

d

diameter

m

e

e-function

unit vector

f

auxiliary variable, function, matrix, column vector

g

auxiliary variable, function, matrix

h

element length, distance, height

m

i

control variable

i

current

A

j

control variable

j

imaginary unit

k, k

constant, complex constant

l

length

m

l

matrix

m

control variable

m

mass

kg

n

normal, number of partial intervals

p

impulse

kg m/s

p

variable, function

r

radius

m

s

constant

s

distance, length

m

t

time

s

u

function, interpolation, approach function

u

voltage

V

û0

voltage amplitude

V

v

function, interpolation, approach function

v

speed

m/s

w

weight, weighting, test, shape function

x

coordinate, path

m

y

coordinate, path

m

y

function

z

coordinate, path

m

Γ

edge of the FEM area

∆

delta, differential

Θ

magnetomotive force

A

Φ

magnetic flux

Vs

Ψ

chained magnetic flux

Vs

Ω

area, sub-area, element

m2

α

coefficient

β

coefficient

γ

Coefficient, boundary value

δ

decay constant

ε

permittivity

As/(Vm)

ε0

permittivity of the vacuum [8, 8542 10−12As/(Vm)]

As/(Vm)

υ

temperature

°C

κ

specific electrical conductivity

m/(Ωmm2)

λ

thermal conductivity

W/(mK)

λ

eigenvalue, Lagrange multiplier

μ

permeability

V s/(Am)

μ0

permeability of the vacuum [4π10−7V s/(Am)]

V s/(Am)

ρ

density

kg/m3

ρ

volume charge density

As/m3

τ

time constant

s

υh

approach, test function

φ

potential

V

φ

interpolation, approach function, angle

φ

angle

rad

ϕ

development, base, triangular function

ω

angular velocity, angular frequency

1/s

∆A, ∆A′

differential surface elements

m2

∆x, ∆y

differential line elements

m

dA

infinitesimal surface element

m2

dx, dy

infinitesimal line elements

m

linear operator

linear operator

zero operator

identity operator

∇

Nabla operator

∆

Delta operator

Chapter 1

Required mathematical basics

„Last time I asked: What does mathematics mean to you?, and some people answered: The manipulation of numbers, the manipulation of structures. And if I had asked what music means to you, would you have answered: The manipulation of notes?“

(Serge Lang, French-American mathematician, 1927-2005) from „The beauty of doing Mathematics“. Serge Lang became known for his work on algebraic geometry and number theory and as the author of many textbooks.

The basics required for the numerical solution of differential equations have been compiled in this chapter. These essentially include matrices, definitions and classifications of differential equations as well as initial and boundary value problems and vector operators. Particularly recommended literature for this are [4], [60] and [67].

1.1Logarithm

The logarithm of x (numerus, logarithmand) to the base a is the real number b (exponent), for which the following applies

The logarithm to the base 10 is called the decadic or Briggsian logarithm. It follows

log10x=lg x

and it applies

log (x · 10α)=α + log x.

Examples of this are

•Example 1:

•Example 2:

Furthermore

log aα + log m

with the numerus or logarithm a, mantissa m and α the index of the logarithm, equal to the exponent of the place value of the first significant digit of the numerus. See also [1], S. 56. In summary, some more useful logarithmic laws are

•Multiplication of the independent parameters

loga(u · v)=loga u + loga v

•Division of the independent parameters

•Exponentiation of the independent variable

loga uvv loga u

•Squaring of the independent variable

1.2Matrices

The matrix notation summarises the calculations with functions and thus increases the overview. For this purpose, a vector operator summarises derivatives. These are marked with a simple symbol (Nabla or Laplace operator). The matrix notation (matrix equations) enables the numerical solution of linear systems of equations by means of the solution methods known in the literature. Therefore, matrix and matrices receive special attention. Selected matrix operations are presented here. These include the necessary matrix calculation rules, the inversion, multiplication of a matrix, matrix textures as well as determinant calculation rules, and much more. Recommended literature is [60], p. 268 ff. and [29], p. 12 ff. (Random matrices – new universal laws).

1.2.1Arithmetic operations with matrices

Table 1.1 summarises the most important algebraic axioms.

Table 1.1: Summary of the most important calculation rules

Note that matrix multiplication is not commutative, which is

A · B≠B · A.

1.2.2Addition and subtraction of two matrices

Two matrices A and B of the same type are added or subtracted by adding or subtracting their corresponding elements

1.2.3Multiplication of a matrix with a scalar

The multiplication of a matrix A with the scalar λ is done by multiplying each individual matrix element with the scalar

1.2.4Square matrix

Examples of square matrices are the diagonal matrices, the symmetrical matrices, normal matrices, Hermitian matrices and the unit matrices.

1.2.5Identity matrix

The identity matrix or unit matrix E is a diagonal matrix in which all elements outside the main diagonal disappear

1.2.6Determinant

A determinant is multiplied by a scalar λ by multiplying the elements of a single line by the scalar

The 3-row determinant is determined according to the rule of Sarrus

det Aa11a22a33 + a12a23a31 + a13a21a32 − a13a22a31 − a11a23a32 − a12a21a33.

A determinant takes the value zero if

•all elements are equal to zero,

•two rows or columns are equal to each other,

•two rows or columns are proportional to each other,

•one row or column is representable as a linear combination of the remaining rows or columns.

An example of this is

the determinant of Dürer’s square from his copper engraving MELENCOLIA I.

1.2.7Subdeterminant or minor

If m arbitrary rows and m arbitrary columns are deleted from an n-row determinant, the result is an (n − m)-row determinant, which is called a subdeterminant (n − m)’th order or minor. An example of this is the determinant A, whose minor M1,2 is sought. This is obtained by deleting the first row and second column

Subdeterminants are required, for example, to calculate the inverse matrix and form the preliminary stage for calculating the adjoints.

1.2.8Adjuncts or algebraic complement

The adjoints, adjuncts or algebraic complement Aadj is formed by subdeterminant formation of the matrix A according to the procedure shown in fig. 1.1. A subsequent multiplication of the elements with the sign (−1)i+k, the i-th row and k-th column, which is shown in fig. 1.1 and transposing leads to the adjoints Aadj of the matrix A.

Figure 1.1: Procedure for the development of the adjuncts

An example of this is

The adjoint must not be confused with the adjoint matrix. The Latin term „adjuncts“ means the subdeterminant assigned to an element of a determinant, where „adjuncts“ means to assign, to attach. The Latin word „complement“ means addition. Adjoints can be used to calculate the inverse of a square matrix.

1.2.9Inverse matrix

The calculation of the inverse matrix A−1

is done using the adjuncts. Furthermore

An example of this is

with

det A=93

and

The inversion of a matrix enables, for example, the solution of linear systems of equations.

1.2.10Transposed of a matrix

For example, transposing a matrix is part of calculating adjoints, or is applied to calculate eigenvalues.

1.2.11Complex conjugate matrix

The conjugate complex number of

za + bi

is

z*a − bi.

The complex conjugate matrix of A is A* in which each element of the matrix is replaced by its complex conjugate element. An example of this is

Swapping the sign of the imaginary unit corresponds to mirroring the imaginary part on the real axis.

1.2.12Hermite conjugate matrix

The Hermitian conjugate matrix or adjoint of a matrix or adjoints of the matrix A of type (m, n) with complex elements is the transpose of its complex conjugate, or the complex conjugate of its transpose

An example of this is

1.2.13Hermitian matrix – self-adjoint matrix

The Hermitian matrix A is a square matrix with complex elements equal to its adjoint matrix

In the case of real element occupation, the notions of symmetric and Hermitian matrices correspond to each other. An example is

Hermitian matrices are used, for example, in systems of linear equations. The Marix was named after Charles Hermite, a French mathematician (1822-1901).

1.2.14Orthogonal matrix

A square matrix A is said to be orthogonal if its transpose is equal to its inverse

ATA−1

or the multiplication of the transposed orthogonal matrix with the orthogonal matrix is equal to the unit matix

ATAE.

An example of this is

so that

Furthermore

is given. Orthogonal matrices are used in systems of linear equations and in matrix decomposition.

1.2.15Unitary matrix

A square matrix A with complex elements is defined as a unitary matrix, if

is. It is thus the transpose of its complex conjugate, which corresponds to the inverted matrix. In the real, the terms unitary and orthogonal coincide. An example of this is

Unitary matrices are used in matrix decomposition.

1.2.16Normal matrix

A square matrix is called a normal matrix if it satisfies the equation

AATATA.

Hermitian, unitary, symmetric and orthogonal matrices are examples of normal matrices. An example of a normal matrix is

1.2.17Norm of a matrix

Given the matrix A with

whose norm with

is calculated. Matrix norms are often used in linear algebra and numerical mathematics. Furthermore, they are used to investigate the convergence of power series of matrices.

1.2.18Conditioned matrix equation and condition number

When solving a matrix equation, numerical problems may arise which need to be evaluated. Given is the matrix equation

•a small change of I

•a small change in I a small change in D, the system is said to be well-conditioned.

The evaluation of a matrix A is done with its condition number cond‖A‖ including its inverse. Here

•cond‖A‖ ≈ 1: well conditioned matrix,

•cond‖A‖ > 1: ill conditioned matrix.

Given are the matrices A and A−1 with

The condition number cond‖A‖ of the matrix A is compared with the maximum sum of the elements of a row

calculated. The matrix is considered ill-conditioned. Furthermore, by means of

calculates the number of decimals (decimal places) that are lost in precision. There is no clear definition here, so care should be taken when using it.

1.2.19Eigenvalue, eigenvector

As an example, consider the matrix equation

where the column vector of the left half of the equation does not match the result vector of the right half of the equation. By changing the left column vector and multiplying it by the matrix again, it follows that

a result vector which is equal to the left column vector. The matrix equation takes the general form

where A is the matrix, is the eigenvector and λ is the scalar eigenvalue. The left-hand side of the equation is a matrix-vector multiplication and the right-hand side of the equation is a scalar multiplication. If in the progression λ is used

is described with the help of the unit matrix E, then the matrix equation follows again in general form

By rearranging it follows

Values for λ are sought which satisfy the equation. The condition is calculated with the characteristic polynomial P(λ)

which arises through the development of the determinant. The determination of eigenvalues is preferably used in physical-technical systems for the calculation of resonance frequencies.

1.2.20Square matrices – a summary

Quadratic matrices of the type (m, m) or Amm are often used to describe physical phenomena and are significant in physics. fig. 1.2 shows a summary.

Figure 1.2: Summary of selected types of quadratic (m, m) matrices

1.3Integral, differential equations

Many processes in science and technology are described by means of differential equations (DEs). In order to facilitate access to differential equations, they are presented here. After initial definitions of terms, a classification of differential equations is given. Furthermore, initial value tasks and boundary value tasks are presented. In the following summary, particular use was made of the literature [4], [62], [68].

1.3.1Definitions

•The variable x is called independent variable or argument of the function y. The variable y is called dependent variable.

•Differential equation (DE) is called an equation in which, in addition to one or more independent variables and one or more functions of these variables, the derivative of this function with respect to the independent variables also occur. The order of a differential equation is equal to the order of the highest derivative occurring in it.

•Partial differential equations (PDEs) contain partial derivatives of a function of several variables.

•A differential equation is called linear if the function and its derivatives occur only linearly, i.e. to the first power.

•A differential equation is called homogeneous if the sum of all terms containing the function f or its derivative of f is equal to zero. Otherwise it is called inhomogeneous.

In tab. 1.2 are examples of differential equations.

Table 1.2: Examples for the representation and naming of differential equations