Essential mathematics for studies in physics

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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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Published 01 October 2012
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F

M

)URP DQJOH WR ˃HOG

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.

Already published

Michel FONKOU, Règles, techniques et pratiques de la
rédaction administrative,2012.
Alexis NGATCHA,Les devoirs à la maison. Réflexion autour
des écoliers africains, 2012.
Gabriel OHANDZA NGONO (éd.),L’épreuve de texte au
cycle d’orientation au Cameroun, 2012.
Lucas PONY,Éthique et développement et économie générale
pour BTS 2. Annales BTS et épreuves corrigées, 2012.
Théophile MBANG, Thierry Stéphane NDEM MBANG,La
chimie dans les grandes écoles et classes scientifiques
préparatoires, 2011.
Emire MAGA MONDESIR, Eliezer MANGUELLE
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
diffusion.harmattan@wanadoo.fr
harmattan1@wanadoo.fr
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 .. . . . . . . . . . . . . . . . . . . . .80
1.4.3 EXPONENTIALFUNCTION .. . . . . . . . . . . . . . .87
1.4.4 HYPERBOLICFUNCTIONS (chx, shx, thx). . . . . . . .88
1.4.5 LOGARITHMFUNCTION .. . . . . . . . . . . . . . . .89
1.4.6 SINUSOIDALFUNCTIONS . . . . . . . . . . . . . . . . .90
1.4.7 TANGENTFUNCTION .. . . . . . . . . . . . . . . . . .91
1.4.8 INVERSETRIGONOMETRIC FUNCTIONS AND INVERSE
HYPERBOLIC FUNCTIONS .. . . . . . . . . . . . . . . .93
1.4.9 COMPLEXFORM OF SINUSOIDAL FUNCTIONS
(illustration 33, page 193) .. . . . . . . . . . . . . . . . . . . . .94

DIFFERENTIALS
2.1 DIFFERENTIALOF A VARIABLE. . . . . . . . . . . . . . . .
2.2 DIFFERENTIALOF A FUNCTION (illustration , page 25,186; 27
page 188). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 ASSIMILATIONOFΔyTOdy. . . . . . . . . . . . . . .
2.2.2 COMPUTATIONRULE .. . . . . . . . . . . . . . . . . . .
2.3 DIFFERENTIALSOF HIGHER ORDER. . . . . . . . . . . . . .
2.4 DIFFERENTIALOF A FUNCTION OF
SEVERAL VARIABLES. . . . . . . . . . . . . . . . . . . . . . .
2.4.1 PARTIALDERIVATIVES .. . . . . . . . . . . . . . . . .
2.4.2 SECONDPARTIAL DERIVATIVE. . . . . . . . . . . . .
2.4.3 DIFFERENTIALOF THE FUNCTIONh=f(x, y, z). . .
2.4.4 IMPLICITFUNCTIONS .. . . . . . . . . . . . . . . . . .
2.4.5 PARAMETRICFUNCTIONS (illustration 24, page 185).
2.5 APPLICATIONTO ERROR CALCULUS. . . . . . . . . . . . .
2.5.1 ABSOLUTEERROR, ABSOLUTE UNCERTAINTY. . .
2.5.2 RELATIVEUNCERTAINTY .. . . . . . . . . . . . . . .
2.5.3 RULESFOR ERROR CALCULATION. . . . . . . . . . .

97
97

98
98
98
99

99
99
100
100
101
101
101
102
102
102

CONTENTS

3

4

5

ix

SERIES EXPANSION105
3.1 ROLLE’STHEOREM .. . . . . . . . . . . . . . . . . . . . . . . .105
3.2 MEANVALUE THEOREM. . . . . . . . . . . . . . . . . . . . .106
3.3 TAYLOR-MACLAURINFORMULA .. . . . . . . . . . . . . . .106
3.3.1 LIMITEDEXPANSION . . . . . . . . . . . . . . . . . . . .107

INTEGRATION 109
4.1 INTEGRALSOF FUNCTION OF ONE VARIABLE. . . . . . .109
4.1.1 PRIMITIVE. . . . . . . . . . . . . . . . . . . . . . . . . .109
4.1.2 DEFINITEINTEGRAL . . . . . . . . . . . . . . . . . . . .110
4.1.3 GEOMETRICINTERPRETATION .. . . . . . . . . . . .110
4.1.4 EXAMPLESOF PRIMITIVES OF FUNCTIONS OF REAL
VARIABLE . . . . . . . . . . . . . . . . . . . . . . . . . . .112
4.1.5 CALCULATIONMETHODS OF INTEGRAL. . . . . . .112
4.1.6 MEANVALUE THEOREM (illustration 33, page 193). .114

4.1.7 DERIVATIONUNDER THE INTEGRAL SIGN. . . . .114
4.2 MULTIPLEINTEGRALS .. . . . . . . . . . . . . . . . . . . . .115
4.2.1 DOUBLEINTEGRAL .. . . . . . . . . . . . . . . . . . . .115
4.2.2 DENSITYDISTRIBUTION - SURFACE INTEGRAL. .115
4.2.3 TRIPLEINTEGRAL .. . . . . . . . . . . . . . . . . . . .116
4.2.4 CURVILINEARINTEGRAL (illustration 19, page 180; 29,
page 189). . . . . . . . . . . . . . . . . . . . . . . . . . . .117
4.2.5 GREEN-RIEMANNFORMULA .. . . . . . . . . . . . . .117
4.3 LENGTH-AREA-VOLUME. . . . . . . . . . . . . . . . . . . . . .118
4.3.1 CALCULATIONOF AN ARC LENGTH OF CURVE. . .118
4.3.2 CALCULATIONOF AREAS. . . . . . . . . . . . . . . .120
4.3.3 CALCULATIONOF VOLUMES. . . . . . . . . . . . . . .123

DIFFERENTIAL EQUATIONS
5.1 GENERALITYAND DEFINITIONS. . . . . . . . . . . . . . . .
5.1.1 GENERALSOLUTION . . . . . . . . . . . . . . . . . . . .
5.1.2 VARIOUSTYPES OF EQUATION. . . . . . . . . . . . .
5.2 FIRSTORDER DIFFERENTIAL EQUATION .. . . . . . . . . .
5.2.1 EQUATIONWHERE THE LEFT SIDE IS A TOTAL
DIFFERENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 FIRSTORDER SEPARABLE EQUATIONS. . . . . . . .

127
127
127
128
128

128
129

205

165

5.5

x

5.4

5.2.3 FIRSTORDER HOMOGENEOUS EQUATIONS. . . . .129
5.2.4 FIRSTORDER DIFFERENTIAL EQUATIONS
(illustration 36, page 195). . . . . . . . . . . . . . . . . . . . . . .130
5.2.5 BERNOUILLIEQUATION . . . . . . . . . . . . . . . . . .132
SECOND ORDER DIFFERENTIAL EQUATIONS. . . . . . . .133
5.3.1 EQUATIONEQUIVALENT TO A FIRST ORDER
EQUATION (illustration 37, page 196). . . . . . . . . . . . . . .133
5.3.2 SECONDORDER LINEAR EQUATION. . . . . . . . . .134
SYSTEM OF FIRST ORDER LINEAR DIFFERENTIAL
EQUATIONS .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139
5.4.1 GENERALCASE .. . . . . . . . . . . . . . . . . . . . . .140
5.4.2 FIRSTORDER LINEAR DIFFERENTIAL EQUATIONS
WITH CONSTANTS COEFFICIENTS. . . . . . . . . . .140
DIFFERENTIAL EQUATION OF ARBITRARY ORDER. . . .142
5.5.1 REDUCTIONOF THE ORDER. . . . . . . . . . . . . . .142
5.5.2 LINEARDIFFERENTIAL EQUATION OF
ORDER n .. . . . . . . . . . . . . . . . . . . . . . . . . . .143
5.5.3 LINEARDIFFERENTIAL EQUATION ASM WITH
CONSTANT COEFFICIENT .. . . . . . . . . . . . . . . . . . .144
5.5.4 OPERATIONALCALCULUS .. . . . . . . . . . . . . . .148
PARTIAL DIFFERENTIAL EQUATION. . . . . . . . . . . . . .151
5.6.1 FIRSTORDER PARTIAL DIFFERENTIAL EQUATIONS151
5.6.2 SECONDORDER LINEAR PARTIAL DIFFERENTIAL
EQUATIONS (IN TWO VARIABLES). . . . . . . . . . .155
5.6.3 EXAMPLEOF SOLUTION FOR THE WAVE EQUATION157
5.6.4 EXAMPLEOF SOLUTION FOR LAPLACE EQUATION158

5.3

IV

APPENDIX

5.6

ILLUSTRATIONS

APPENDIX A

V

Illustrations

CONTENTS

163

203

CONTENTS

xi

A FORSYMMETRIC GROUNDS205
A.1 SYMMETRICCURIE’S LAW. . . . . . . . . . . . . . . . . . . .205
A.2 INVARIANCEPRINCIPLE OF LAWS BY SYMMETRIC
TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206
A.3 EXAMPLEOF PRACTICAL USE OF SYMMETRIES. . . . . .206
A.3.1 ELECTRICALFIELD CREATED AT A POINT M BY A
VOLUMIC DISTRIBUTION OF CHARGES (fig.1). . . .206
A.3.2 CENTEROF INERTIA-MATRIX OF INERTIA FOR AN
HOMOGENEOUS RIGHT CONE OF AXISOZ, OF HEIGHT
HAND SUMMITO(fig. 2).. . . . . . . . . . . . . . . . .208

B

DIMENSIONS-HOMOGENEITY

211

APPENDIX B211
B.1 MEASUREMENT .. . . . . . . . . . . . . . . . . . . . . . . . . .211
B.2 SYSTEMOF UNITS .. . . . . . . . . . . . . . . . . . . . . . . . .211
B.3 INTERNATIONALSYSTEM .. . . . . . . . . . . . . . . . . . .212
B.4 DIMENSIONS .. . . . . . . . . . . . . . . . . . . . . . . . . . . .212
B.5 HOMOGENEITY. . . . . . . . . . . . . . . . . . . . . . . . . . .212

INDEX

215

PREFACE

Mathematics has always been an essential tool for the physicists.It provides an
appropriate language and manner of reasoning for the physical
sciences.Mathematics aids in the formulation of physical problems.This makes a fundamental
mathematical background an essential requirement for the physics student.The
authors designed this book to meet the needs of the undergraduate physics student.
Though not meant to give a formal mathematics course, the book can be
considered an essential tool for both undergraduate and graduate students.It is a good
reference book for the physics lecturer, and even the mathematics students.The
mathematical concepts presented in this book will find their application in many
different areas of physics including mechanics, electrodynamics, electromagnetic
theory, etc.Each topic is treated through a combination of definitions, principles,
theorems, laws and illustrations.The illustrations are however presented at the
end. Inaddition an appendix is presented containing further information
considered equally useful for the study of physics.Topics covered in the book include
geometry, vectors, matrix transformation, integration and differential equations.
Reading through this book will stimulate the readerŠs interest in the subject and
certainly cause physics students to enjoy the study of physics.

.
Tabod Charles Tabod, July 2012
Associate Professor of Geophysics,
University of Yaounde I,
Cameroon
and
Vice Dean in charge of Academic Affairs
University of Bamenda,
Cameroon

FOREWORD

We are Lecturers at the Faculty of Science of the University of Yaoundé
ICameroon.
After investigating the reasons for the failure of the first year physics students, we
realized that they generally lacked the required mathematical background.So we
had idea to propose a hand book covering most of the basic mathematics necessary
for the understanding of the physics course but which is not usually taught in the
mathematics course.
Throughout the chapters, there is a large numbers of examples and some
illustrations in physics are given at the end of the book.
Although more modern methods of analysis exist nowadays, our students do not
have easy access to these tools.They therefore have to master all the methods
even the most empirical ones in order to be acquainted with the universality of
science.
They can also consult other books with many exercises like that by Michel Hulin
and Marie Francoise Quentin in the "Collection U".
We have not mentioned random aspects in this book because abundant literature
exists on this and also because these notions are mainly useful for quantum and
statistical physics at higher level.
Finally we wish that our young students understand that mathematical ideals are
useful in the description of physical realities and therefore we expect this book to
inspire them with more motivations for physics.
. Theauthors

Part I

GEOMETRY AND
TRIGONOMETRY

Chapter 1

SOME USUAL GEOMETRIC
SHAPES

1.1 AREAS

4

1.2

CHAPTER 1.SOME USUAL GEOMETRIC SHAPES

VOLUMES

Chapter

BASIC

2.1

2.1.1

2

GEOMETRY

THE ANGLE

DEFINITIONS

PLANE ANGLE: It isa figure formed with two straight lines called sides that meet
at a point called summit(fig.1). Theangle is an important position parameter in
physics.

The symbolxOyis read anglexOy

O

Figure 1

y

x

EQUAL ANGLES:Two angles are said to be equal if they are superposable.

Bisector: itis a straightozline that divides a given angle into two equal
angles from the summit(fig.2).

6

O


CHAPTER 2.

Figure 2

xoz=zoy

y

x

BASIC GEOMETRY

z

SPECIAL ANGLES
STRAIGHT ANGLE:Its two sides are on the same straight line (fig.3).
z

y

Figure 3

O

x

(2.1.1)

RIGHT ANGLE: The bisectorozdivides a straight angle into two right angles
(fig. 3).
ozis said to be perpendicular toox(or tooy)
O
ACUTE ANGLE:Such an angle is smaller than right angle.
OBTUSE ANGLE:An obtuse angle is greater than the right angle.

2.1.2

ANGLE UNITS

THE DEGREE:The value of a plane angle is generally measured in degree and

with the help of a protractor.Its symbol is. Thesubunits are:

minute of angle (’);

second of angle (");


1 =60’

1’ = 60"

2.1.

THE ANGLE

7

◦ ◦
Examples :The value of a right angle is 90and that of a straight angle is 180.
th
The degree can also be considered as the90part of a right angle.
THE GRADE or GON :Its symbol is (gr).The grade is mostly used in
gerth
manic countries.It is the100The centigrade is apart of the right angle.
subunit.
THE RADIAN: Thisis a very convenient unit in physics, especially when
dealing with circles.The radian (rd) is the measure of the central angle which
subtends an arc length equal to the radius of a circle and it is independent of the
size of this circle(fig.4).
A

O

α
R


B

Figure 4

Generally, the length of an arcABis the product of the radius by the central
angle given in radian which subtends it.


arc AB=Rα(αin rd)

In particular, ifarc AB=R, thenα= 1rd

The central angle which subtends a half circle measuresπradians

The circumference of a circle with radiusRis2πR.

(2.1.2)

Relationship between remarkable angles

Straight angle== 180πrd= 200gr
◦π
Right angle= 90=rd= 100gr
2
◦π
60 =rd= 66.66gr
3
◦π
45 =rd= 50gr
4
◦π
30 =rd= 33.33gr
6
•Supplementary anglesangles are said to be supplementary if the: Two
sum of their values is equal toπradians.

Complementary angles: Twoangles are said to be complementary if the
π
sum of their values is equal toradians.
2

8

2.1.3

CHAPTER 2.

CHARACTERISTIC PROPERTIES

BASIC GEOMETRY

Angles with parallel sides (illustration 1, page 165 ) (fig.5)

O

Then we have

Y

Figure 5

X

X”

Y’

O’

′ ′′ ′
OX//O XandOY //OY


′ ′ ′′ ′′′
XOY=X O YorXOY+XY O=π

Special case of equal angles(fig. 6)


- Alternate angles (ex :BIDandCJ A)


- Corresponding angles (CIAandCJ A)
- Supplementary angles :


′ ′
AIC+A JD=πandBID+IJ B=π

A

A’

C

I

Figure 6

Angles of a triangle( illustration 2, page 165)(fig.7).

J

D

X’

B

B’

(2.1.3)

(2.1.4)

2.1.

THE ANGLE

B


1.BAC+CAD=π

A✲

D

Figure 7

C

X


2.AX//BCalorsDAX=DBCandCAX=ACBthen

ABC+ACB=CAD

Consequence


ABC+BCA+CAB=π

Summing all angles in a triangle gives the valueπ
Angles with perpendicular sides


- Two acute angles:They are equalBAD=DCX

- An acute and an obtuse angles are supplementary:BAD+BCD=π

B

Inscribed angles in a circle:

A

C
Figure 8

X

D

9

(2.1.5)

Definition 2.1.1.: Theinscribed angle has it summit on the circle and subtends

an arc length such asBAC(fig.9)