Essential mathematics for studies in physics

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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Publié par
Ajouté le 01 octobre 2012
Nombre de lectures 8
EAN13 9782296505230
Langue English
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Maga Mondésir EmireEssential mathematics
Manguelle Dicoum Eliézerfor studies in physics
Mbianda Gilbert
For undergraduate students
From angle to feld
Essential mathematics Tis book is a broad spectrum summary of essential mathematics for
undergraduate physics students at levels I, II, and III and beyond. Basic
geometry often forgotten is reviewed. Mathematical aspects that are new for for studies in physics
beginners but which are often not the focus of physics lecture are explained.
Tis book contains many examples.
Some of the chapters may be very useful to mathematics students. With this For undergraduate students
book, the students will be in more comfortable, confdent and interested in the
study of physics. From angle to feld
Dr. Maga Mondésir Emire is a Lecturer at University of Yaoundé I-Cameroon.
eShe got her Doctorate Diploma 3 cycle at the University Pierre et Marie Curie
(Paris 6) in Mathematical Physics.
Pr. Manguel le Dicoum is a Full Professor in geophysics at University of Yaoundé
eI-Cameroon. Docteur es-Sciences, he obtained a Doctorate Diploma 3 cycle at the
University Pierre et Marie Curie (Paris 6) in high energy physics.
Dr. Mbianda Gilbert is a Lecturer at University of Yaoundé I-Cameroon. He got
ehis Doctorate Diploma 3 cycle at the University Pierre et Marie Curie (Paris 6). He
is also an engenieer from ENSAE (Supaero) of Toulouse-France.
Preface by Tabod Charles Tabod
ISBN : 978-2-336-00284-2
Cours & Manuels25 e
Maga Mondésir Emire
Essential mathematics for studies in physics
Manguelle Dicoum Eliézer
For undergraduate students - From angle to feld
Mbianda GilbertESSENTIAL MATHEMATICS
FOR STUDIES IN PHYSICS
For undergraduate students
From angle to fieldCollection « Cours et Manuels »
Harmattan Cameroun

Under the supervision of Roger MONDOUE
and Eric Richard NYITOUEK AMVENE

Most African students complete their studies without
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It is therefore necessary to publish and to promote these
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MAGA MOND ÉSIR Emire
MANGUELLE DICOUM Eliézer
MBIANDA Gilbert







ESSENTIAL MATHEMATICS
FOR STUDIES IN PHYSICS

For undergraduate students
From angle to field


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 of Charles 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 xiii
I GEOMETRY AND TRIGONOMETRY 1
1 SOME USUAL GEOMETRIC SHAPES 3
1.1 AREAS ................................. 3
1.2 VOLUMES ............................... 4
2 BASIC GEOMETRY 5
2.1 THE ANGLE .............................. 5
2.1.1 DEFINITIONS ......................... 5
2.1.2 ANGLE UNITS ........................ 6
2.1.3 CHARACTERISTIC PROPERTIES ............. 8
2.2 TRIANGLE 10
2.2.1 DEFINITIONS 10
2.2.2 REMARKABLE LINES OF A TRIANGLE ........ 11
2.2.3 PROPERTIES ......................... 12
2.3 SPECIAL TRIANGLES ........................ 15
3 BASIC TRIGONOMETRY 17
3.1 DIRECTED ANGLE.......................... 17
3.2 TRIGONOMETRIC CIRCLE..................... 18
3.3 APPLICATION TO TRIANGLE ................... 19
3.3.1 SCALENE .................... 19
3.3.2 RIGHT TRIANGLE,( Fig. 4),(illustration 5 page 169) . . . 19
3.4 TRIGONOMETRIC FORMULAE .................. 21vi CONTENTS
II VECTORS AND MATRICES CALCULATION 23
1 25
1.1 LOCATION IN SPACE ........................ 25
1.1.1 INTRODUCTION ....................... 25
1.1.2 LOCATION ON A STRAIGHT LINE ............ 26
1.1.3 LOCATION IN A PLANE .................. 26
1.1.4 LOCATION IN SPACE.................... 27
1.2 VECTORS ............................... 28
1.2.1 DEFINITIONS ......................... 28
1.2.2 VECTOR SPACE ....................... 29
1.2.3 OPERATIONS ON VECTORS................ 30
2 LINEAR TRANSFORMATIONS AND MATRICES 37
2.1 REMIND OF BASIC PROPERTIES OF TRANSFORMATIONS . 37
2.1.1 DEFINITIONS 37
2.1.2 EXAMPLES .......................... 38
2.2 MATRICES ASSOCIATED TO A LINEAR TRANSFORMATION
IN VECTOR SPACE E ........................ 38
2.2.1 DEFINITION 38
2.2.2 EXAMPLES 39
2.3 MATRIX OPERATIONS ....................... 43
2.3.1 EQUALITY OF TWO MATRICES ............. 43
2.3.2 ZERO MATRIX 44
2.3.3 OPPOSITE MATRIX ..................... 44
2.3.4 ADDITION OF TWO MATRICES WITH THE SAME DI-
MENSION ........................... 44
2.3.5 PRODUCT OF TWO MATRICES A(n,p) and B(n,p ).. 44
2.3.6 MULTIPLICATION BY A SCALAR............. 44
2.3.7 TRANSPOSED MATRIX ................... 45
2.3.8 INVERSE OF A SQUARE MATRIX ............ 45
2.3.9 EIGENVALUES AND EIGENVECTORS ......... 45
2.4 CHANGE OF BASIS ......................... 46
3 VECTOR DERIVATION 49
3.1 ORDINARY DERIVATIVES OF VECTORS ............ 49CONTENTS vii
3.2 CURVES IN 3 DIMENSIONAL SPACE ............... 50
3.3 DERIVATION FORMULAE ..................... 51
3.4 PARTIAL DERIVATIVES OF VECTORS.............. 51
3.5 DIFFERENTIALS OF VECTORS . ................. 52
4 NOTION OF FIELD 53
4.1 INTRODUCTION ........................... 53
4.2 DEFINITIONS ............................. 54
4.2.1 THE FIELD .......................... 54
4.2.2 FIELD LINES, TUBE OF FIELD .............. 54
4.2.3 TUBE OF FIELD ....................... 55
4.3 OPERATIONS OF FIELD ...................... 55
4.3.1 THE CIRCULATION OF A VECTOR FIELD ....... 55
4.3.2 THE FLUX OF VECTOR A FIELD............. 56
4.3.3 THE DIFFERENTIAL VECTOR OPERATOR NABLA ∇ 56
5 ORTHOGONAL CURVILINEAR COORDINATES 63
5.1 INTRODUCTION ........................... 63
5.2 ELEMENTARY ARCS AND ELEMENTARY VOLUMES..... 64
5.3 EXPRESSION OF FIELD OPERATORS ................ 65
5.4 SPECIAL SYSTEM OF CURVILINEAR COORDINATES .... 66
5.4.1 CYLINDRICAL COORDINATES .............. 66
5.4.2 SPHERICALTES ............... 67
III FONCTIONS AND INTEGRATION 69
1 COMMON FUNCTIONS 71
1.1 DERIVATION (illustration 24, page 185) .............. 71
1.1.1 DEFINITION OF DERIVATIVE AT A POINT ...... 71
1.1.2 GEOMETRICAL INTERPRETATION ........... 71
1.1.3 DERIVATIVE AS A FUNCTION 72
1.2 THE USE OF DERIVATIVES IN THE STUDY OF FUNCTIONS 73
1.2.1 FIRSTATIVE ..................... 73
1.2.2 SECOND DERIVATIVE ................... 74
1.3 GENERAL RULE TO STUDY FUNCTIONS............ 75
1.3.1 THE DOMAIN OF DEFINITION: .............. 75viii CONTENTS
1.3.2 NATURE OF THE FUNCTION (EVEN, ODD, PERIODIC) 75
1.3.3 ASYMPTOTE ......................... 76
1.4 COMMON FUNCTIONS ....................... 79
1.4.1 LINEAR .................... 79
1.4.2 CONIC SECTIONS ...................... 80
1.4.3 EXPONENTIAL FUNCTION ................ 87
1.4.4 HYPERBOLIC FUNCTIONS (chx, shx, thx) ........ 88
1.4.5 LOGARITHM FUNCTION ................. 89
1.4.6 SINUSOIDAL FUNCTIONS 90
1.4.7 TANGENT FUNCTION ................... 91
1.4.8 INVERSE TRIGONOMETRIC FUNCTIONS AND INVERSE
HYPERBOLIC FUNCTIONS................. 93
1.4.9 COMPLEX FORM OF SINUSOIDAL FUNCTIONS (illus-
tration 33, page 193)...................... 94
2 DIFFERENTIALS 97
2.1 DIFFERENTIAL OF A VARIABLE ................ 97
2.2 OF A FUNCTION (illustration , page 25,186; 27
page 188) ................................ 98
2.2.1 ASSIMILATION OF Δy TO dy ............... 98
2.2.2 COMPUTATION RULE.................... 98
2.3 DIFFERENTIALS OF HIGHER ORDER .............. 99
2.4 DIFFERENTIAL OF A FUNCTION OF
SEVERAL VARIABLES ....................... 99
2.4.1 PARTIAL DERIVATIVES .................. 99
2.4.2 SECOND PARTIAL DERIVATIVE ............. 100
2.4.3 DIFFERENTIAL OF THE FUNCTION h = f(x,y,z) ... 100
2.4.4 IMPLICIT FUNCTIONS ................... 101
2.4.5 PARAMETRIC (illustration 24, page 185) . 101
2.5 APPLICATION TO ERROR CALCULUS ............. 101
2.5.1 ABSOLUTE ERROR, ABSOLUTE UNCERTAINTY . . . 102
2.5.2 RELATIVE UNCERTAINTY ................ 102
2.5.3 RULES FOR ERROR CALCULATION ........... 102CONTENTS ix
3 SERIES EXPANSION 105
3.1 ROLLE’S THEOREM ......................... 105
3.2 MEAN VALUE THEOREM ..................... 106
3.3 TAYLOR-MACLAURIN FORMULA ................ 106
3.3.1 LIMITED EXPANSION.................... 107
4 INTEGRATION 109
4.1 INTEGRALS OF FUNCTION OF ONE VARIABLE ....... 109
4.1.1 PRIMITIVE .......................... 109
4.1.2 DEFINITE INTEGRAL .................... 110
4.1.3 GEOMETRIC INTERPRETATION ............. 110
4.1.4 EXAMPLES OF PRIMITIVES OF FUNCTIONS OF REAL
VARIABLE ........................... 112
4.1.5 CALCULATION METHODS OF INTEGRAL ....... 112
4.1.6 MEAN VALUE THEOREM (illustration 33, page 193) . . 114

4.1.7 DERIVATION UNDER THE INTEGRAL SIGN ..... 114
4.2 MULTIPLE INTEGRALS ...................... 115
4.2.1 DOUBLE INTEGRAL..................... 115
4.2.2 DENSITY DISTRIBUTION - SURFACE INTEGRAL . . 115
4.2.3 TRIPLE INTEGRAL 116
4.2.4 CURVILINEAR INTEGRAL (illustration 19, page 180; 29,
page 189) ............................ 117
4.2.5 GREEN-RIEMANN FORMULA ............... 117
4.3 LENGTH-AREA-VOLUME...................... 118
4.3.1 CALCULATION OF AN ARC LENGTH OF CURVE . . . 118
4.3.2TION OF AREAS . . .............. 120
4.3.3 CALCULATION OF VOLUMES ............... 123
5 DIFFERENTIAL EQUATIONS 127
5.1 GENERALITY AND DEFINITIONS ................ 127
5.1.1 GENERAL SOLUTION .................... 127
5.1.2 VARIOUS TYPES OF EQUATION ............. 128
5.2 FIRST ORDER DIFFERENTIAL EQUATION........... 128
5.2.1 EQUATION WHERE THE LEFT SIDE IS A TOTAL DIF-
FERENTIAL .......................... 128
5.2.2 FIRST ORDER SEPARABLE EQUATIONS ........ 129x CONTENTS
5.2.3 FIRST ORDER HOMOGENEOUS EQUATIONS ..... 129
5.2.4 FIRST DIFFERENTIAL EQUATIONS (illustra-
tion 36, page 195) ....................... 130
5.2.5 BERNOUILLI EQUATION . . . ............... 132
5.3 SECOND ORDER DIFFERENTIAL EQUATIONS ........ 133
5.3.1 EQUATION EQUIVALENT TO A FIRST ORDER EQUA-
TION (illustration 37, page 196) 133
5.3.2 SECOND ORDER LINEAR EQUATION .......... 134
5.4 SYSTEM OF FIRST ORDER LINEAR DIFFERENTIAL EQUA-
TIONS.................................. 139
5.4.1 GENERAL CASE ....................... 140
5.4.2 FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS
WITH CONSTANTS COEFFICIENTS ........... 140
5.5 DIFFERENTIAL EQUATION OF ARBITRARY ORDER .... 142
5.5.1 REDUCTION OF THE ORDER ............... 142
5.5.2 LINEAR DIFFERENTIAL EQUATION OF
ORDER n............................ 143
5.5.3 LINEAR EQUATION ASM WITH CON-
STANT COEFFICIENT.................... 144
5.5.4 OPERATIONAL CALCULUS ................ 148
5.6 PARTIAL DIFFERENTIAL EQUATION .............. 151
5.6.1 FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 151
5.6.2 SECOND ORDER LINEAR PARTIAL DIFFERENTIAL
EQUATIONS (IN TWO VARIABLES) ........... 155
5.6.3 EXAMPLE OF SOLUTION FOR THE WAVE EQUATION 157
5.6.4 OF FOR LAPLACE EQUATION 158
IV ILLUSTRATIONS 163
Illustrations 165
V APPENDIX 203 A 205CONTENTS xi
A FOR SYMMETRIC GROUNDS 205
A.1 CURIE’S LAW .................... 205
A.2 INVARIANCE PRINCIPLE OF LAWS BY SYMMETRIC TRANS-
FORMATIONS ............................. 206
A.3 EXAMPLE OF PRACTICAL USE OF SYMMETRIES ...... 206
A.3.1 ELECTRICAL FIELD CREATED AT A POINT M BY A
VOLUMIC DISTRIBUTION OF CHARGES (fig.1) .... 206
A.3.2 CENTER OF INERTIA-MATRIX OF INERTIA FOR AN
HOMOGENEOUS RIGHT CONE OF AXISOZ, OF HEIGHT
H AND SUMMIT O(fig. 2). ................. 208
B DIMENSIONS-HOMOGENEITY 211
APPENDIX B 211
B.1 MEASUREMENT ........................... 211
B.2 SYSTEM OF UNITS.......................... 211
B.3 INTERNATIONAL SYSTEM .................... 212
B.4 DIMENSIONS ............................. 212
B.5 HOMOGENEITY 212
INDEX 215PREFACE
Mathematics has always been an essential tool for the physicists. It provides an
appropriate language and manner of reasoning for the physical sciences. Mathe-
matics 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 consid-
ered 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. In addition an appendix is presented containing further information consid-
ered 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,
CameroonFOREWORD
We are Lecturers at the Faculty of Science of the University of Yaoundé I-
Cameroon.
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 illustra-
tions 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.
. The authorsPart I
GEOMETRY AND
TRIGONOMETRYChapter 1
SOME USUAL GEOMETRIC
SHAPES
1.1 AREAS4 CHAPTER 1. SOME USUAL GEOMETRIC SHAPES
1.2 VOLUMESChapter 2
BASIC GEOMETRY
2.1 THE ANGLE
2.1.1 DEFINITIONS
PLANE ANGLE: It is a figure formed with two straight lines called sides that meet
at a point called summit(fig. 1). The angle is an important position parameter in
physics.
The symbol xOy is read angle xOy
y

O
x
Figure 1
EQUAL ANGLES: Two angles are said to be equal if they are superposable.
Bisector : it is a straight oz line that divides a given angle into two equal
angles from the summit(fig. 2).6 CHAPTER 2. BASIC GEOMETRY
y
z
O
x
Figure 2
xoz = zoy (2.1.1)
SPECIAL ANGLES
STRAIGHT ANGLE: Its two sides are on the same straight line (fig. 3).
z

xy O
Figure 3
RIGHT ANGLE: The bisector oz divides a straight angle into two right angles
(fig. 3).
oz is said to be perpendicular to ox (or to oy)
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 . The subunits are:
◦• minute of angle (’); 1 = 60’
• second of angle ("); 1’ = 60"2.1. THE ANGLE 7
◦ ◦Examples : The value of a right angle is 90 and that of a straight angle is 180 .
thThe degree can also be considered as the 90 part of a right angle.
THE GRADE or GON : Its symbol is (gr). The grade is mostly used in ger-
thmanic countries. It is the 100 part of the right angle. The centigrade is a
subunit.
THE RADIAN : This is 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

α B
O R
Figure 4

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

arc AB= Rα ( α in rd) (2.1.2)

• In particular, if arc AB= R, then α= 1rd
• The central angle which subtends a half circle measures π radians
• The circumference of a circle with radius R is 2πR.
Relationship between remarkable angles
◦Straight angle = 180 = π rd = 200 gr
◦ πRight angle =90 = rd = 100 gr2
◦ π60 = rd =66.66 gr3
◦ π45 = rd =50 gr4
π◦30 = rd =33.33 gr
6
• Supplementary angles: Two angles are said to be supplementary if the
sum of their values is equal to π radians.
• Complementary angles : Two angles are said to be complementary if the
π
sum of their values is equal to radians.
28 CHAPTER 2. BASIC GEOMETRY
2.1.3 CHARACTERISTIC PROPERTIES
Angles with parallel sides (illustration 1, page 165 ) (fig.5)
Y Y’
X
X’
O’O
X”
Figure 5

OX//O X and OY//O Y
Then we have
XOY = X O Y or XOY +Y O X = π (2.1.3)
Special case of equal angles (fig. 6)
- Alternate angles (ex : BID and CJA )
- Corresponding angles (CIA and CJA )
- Supplementary angles :
AIC +AJD = π and BID +IJB = π (2.1.4)
C
IAB
A’ J B’
D
Figure 6
Angles of a triangle( illustration 2, page 165)(fig. 7).2.1. THE ANGLE 9
D
XA
B C
Figure 7
1. BAC +CAD = π
2. AX//BC alors DAX = DBC and CAX = ACB then
ABC +ACB =CAD
Consequence
ABC +BCA+CAB = π (2.1.5)
Summing all angles in a triangle gives the value π
Angles with perpendicular sides
- Two acute angles: They are equal BAD = DCX
- An acute and an obtuse angles are supplementary: BAD +BCD = π
A
D
B
C X
Figure 8
Inscribed angles in a circle:
Definition 2.1.1. : The inscribed angle has it summit on the circle and subtends
an arc length such as BAC (fig.9)