Essential mathematics for studies in physics

Eliézer Manguelle Dicoum, Emire Maga Mondésir, Gilbert Mbianda

Description

This book is a summary of essential mathematics for undergraduate physics students. Throughout many examples, it developps basic geometry often forgotten or mathematical aspects which are not the focus of physics teachers. It will permit students to feel more comfortable and confident with the study of physics.

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Sujets

• manuels
• Cours
• Annales

Informations

• Publié par: L'Harmattan
• Date de parution: 01 octobre 2012
• Nombre de lectures: 17
• EAN13: 9782296505230
• Langue: Français
• Poids de l'ouvrage: 2 Mo

Extrait

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Maga Mondésir Emire
Manguelle Dicoum Eliézer
Mbianda Gilbert

Essential mathematics
for studies in physics

For undergraduate students
)URP DQJOH WR ˃HOG

)RU XQGHUJUDGXDWH VWXGHQWV )URP DQJOH WR ÀHOG

Preface by Tabod Charles Tabod

Cours & Manuels

ESSENTIAL MATHEMATICS
FOR STUDIES IN PHYSICS

For undergraduate students

From angle to field

Collection « Cours et Manuels »
Harmattan Cameroun

Under the supervision of Roger MONDOUE
and Eric Richard NYITOUEK AMVENE

Most African students complete their studies without
having direct access to primary sources. Their professors'
courses or textbooks are the only educational resources
available.
It is therefore necessary to publish and to promote these
courses and textbooks, in order to grant most students access
to the best education possible.
The Courses & Textbooks series is dedicated to teachers
and professors in any discipline, from elementary school to
middle school, high school or even university, whose main
concern is to improve the education level, and to promote the
development Africa has expected for so long.

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DICOUM, Gilbert MBIANDA,L’indispensable
mathématique pour les études en physique. Premier cycle
universitaire. De l’angle au champ, 2011.

MAGA MONDeSIR Emire
MANGUEL/E DICOUM Elipzer
MBIANDA Gilbert







ESSENTIAL MATHEMATICS
FOR STUDIES IN PHYSICS

For undergraduate students
From angle tofield


Preface by Tabod Charles Tabod







L’Harmattan

Translated from the French version by:
Emire MAGA MONDESIR,Author
Bertrand SITAMTZE YOUMBI,PhD in Material Sciences
(Yaounde I-Cameroon)
Under the supervision ofCharles TABOD TABOD,PhD in
Geophysics (UK),
Associated Professor and Vice-Dean of Academic Affairs in the
Faculty of Science, University of Bamenda-Cameroon.

© L’Harmattan, 2012
5-7, rue de l’École-Polytechnique ; 75005 Paris
http://www.librairieharmattan.com
[e-mail]
[e-mail]
ISBN : 978-2-336-00284-2
EAN : 9782336002842

Contents

FOREWORD

I

1

2

3

GEOMETRY AND TRIGONOMETRY

SOME USUAL GEOMETRIC SHAPES
1.1 AREAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 VOLUMES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BASIC GEOMETRY
2.1 THEANGLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 DEFINITIONS. . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 ANGLEUNITS .. . . . . . . . . . . . . . . . . . . . . . .
2.1.3 CHARACTERISTICPROPERTIES . . . . . . . . . . . . .
2.2 TRIANGLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 DEFINITIONS. . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 REMARKABLELINES OF A TRIANGLE. . . . . . . .
2.2.3 PROPERTIES. . . . . . . . . . . . . . . . . . . . . . . . .
2.3 SPECIALTRIANGLES . . . . . . . . . . . . . . . . . . . . . . . .

BASIC TRIGONOMETRY
3.1 DIRECTEDANGLE .. . . . . . . . . . . . . . . . . . . . . . . . .
3.2 TRIGONOMETRICCIRCLE .. . . . . . . . . . . . . . . . . . . .
3.3 APPLICATIONTO TRIANGLE. . . . . . . . . . . . . . . . . . .
3.3.1 SCALENETRIANGLE .. . . . . . . . . . . . . . . . . . .
3.3.2 RIGHTTRIANGLE,( Fig.4),(illustration 5 page 169) .. .
3.4 TRIGONOMETRICFORMULAE .. . . . . . . . . . . . . . . . .

xiii

1

3
3
4

5
5
5
6
8
10
10
11
12
15

17
17
18
19
19
19
21

vi

II

1

2

3

CONTENTS

VECTORS AND MATRICES CALCULATION

VECTORS
1.1 LOCATIONIN SPACE. . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . .
1.1.2 LOCATIONON A STRAIGHT LINE. . . . . . . . . . . .
1.1.3 LOCATIONIN A PLANE. . . . . . . . . . . . . . . . . .
1.1.4 LOCATIONIN SPACE .. . . . . . . . . . . . . . . . . . .
1.2 VECTORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 DEFINITIONS. . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 VECTORSPACE .. . . . . . . . . . . . . . . . . . . . . .
1.2.3 OPERATIONSON VECTORS .. . . . . . . . . . . . . . .

LINEAR TRANSFORMATIONS AND MATRICES
2.1 REMINDOF BASIC PROPERTIES OF TRANSFORMATIONS .
2.1.1 DEFINITIONS. . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 EXAMPLES. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 MATRICESASSOCIATED TO A LINEAR TRANSFORMATION
IN VECTOR SPACEE. . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 DEFINITION. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 EXAMPLES. . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 MATRIXOPERATIONS .. . . . . . . . . . . . . . . . . . . . . .
2.3.1 EQUALITYOF TWO MATRICES. . . . . . . . . . . . .
2.3.2 ZEROMATRIX . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 OPPOSITEMATRIX .. . . . . . . . . . . . . . . . . . . .
2.3.4 ADDITIONOF TWO MATRICES WITH THE SAME
DIMENSION .. . . . . . . . . . . . . . . . . . . . . . . . . .
′ ′
2.3.5 PRODUCTOF TWO MATRICESA(n, p)andB(n ,p). .
2.3.6 MULTIPLICATIONBY A SCALAR .. . . . . . . . . . . .
2.3.7 TRANSPOSEDMATRIX . . . . . . . . . . . . . . . . . . .
2.3.8 INVERSEOF A SQUARE MATRIX. . . . . . . . . . . .
2.3.9 EIGENVALUESAND EIGENVECTORS. . . . . . . . .
2.4 CHANGEOF BASIS. . . . . . . . . . . . . . . . . . . . . . . . .

VECTOR DERIVATION
3.1 ORDINARYDERIVATIVES OF VECTORS. . . . . . . . . . . .

23

25
25
25
26
26
27
28
28
29
30

37
37
37
38

38
38
39
43
43
44
44

44
44
44
45
45
45
46

49
49

5

FONCTIONS AND INTEGRATION

CONTENTS

NOTION OF FIELD
4.1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 DEFINITIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 THEFIELD .. . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 FIELDLINES, TUBE OF FIELD. . . . . . . . . . . . . .
4.2.3 TUBEOF FIELD. . . . . . . . . . . . . . . . . . . . . . .
4.3 OPERATIONSOF FIELD. . . . . . . . . . . . . . . . . . . . . .
4.3.1 THECIRCULATION OF A VECTOR FIELD. . . . . . .
4.3.2 THEFLUX OF VECTOR A FIELD .. . . . . . . . . . . .

4.3.3 THEDIFFERENTIAL VECTOR OPERATOR NABLA∇

III

1

71
71
71
71
72
73
73
74
75
75

69

63
63
64
65
66
66
67

ORTHOGONAL CURVILINEAR COORDINATES
5.1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 ELEMENTARYARCS AND ELEMENTARY VOLUMES .. . . .
5.3EXPRESSION OF FIELD OPERATORS. . . . . . . . . . . . . . . .
5.4 SPECIALSYSTEM OF CURVILINEAR COORDINATES. . . .
5.4.1 CYLINDRICALCOORDINATES .. . . . . . . . . . . . .
5.4.2 SPHERICALCOORDINATES .. . . . . . . . . . . . . .

50
51
51
52

53
53
54
54
54
55
55
55
56
56

CURVES IN 3 DIMENSIONAL SPACE. . . . . . . . . . . . . . .
DERIVATION FORMULAE. . . . . . . . . . . . . . . . . . . . .
PARTIAL DERIVATIVES OF VECTORS .. . . . . . . . . . . . .
DIFFERENTIALS OF VECTORS. . . . . . . . . . . . . . . . . .

vii

3.2
3.3
3.4
3.5

4

COMMON FUNCTIONS
1.1 DERIVATION(illustration 24, page 185). . . . . . . . . . . . . .
1.1.1 DEFINITIONOF DERIVATIVE AT A POINT. . . . . .
1.1.2 GEOMETRICALINTERPRETATION .. . . . . . . . . .
1.1.3 DERIVATIVEAS A FUNCTION. . . . . . . . . . . . . .
1.2 THEUSE OF DERIVATIVES IN THE STUDY OF FUNCTIONS
1.2.1 FIRSTDERIVATIVE .. . . . . . . . . . . . . . . . . . . .
1.2.2 SECONDDERIVATIVE .. . . . . . . . . . . . . . . . . .
1.3 GENERALRULE TO STUDY FUNCTIONS .. . . . . . . . . . .
1.3.1 THEDOMAIN OF DEFINITION:. . . . . . . . . . . . . .

viii

2

1.4

CONTENTS

1.3.2 NATUREOF THE FUNCTION (EVEN, ODD, PERIODIC)75
1.3.3 ASYMPTOTE. . . . . . . . . . . . . . . . . . . . . . . . .76
COMMON FUNCTIONS. . . . . . . . . . . . . . . . . . . . . . .79
1.4.1 LINEARFUNCTIONS .. . . . . . . . . . . . . . . . . . .79
1.4.2 CONICSECTIONS .. . . . . . . . .