ConciseHandbook-ofMathematics

and PhysicsAlexander G.AIenitsynEugene I. ButikovAlexander S. Kondratyev

CRC Press Nauka PublishersBoca Raton New York Fizmatlit, Moscow

Contents

I MATHEMATICS 1

1 Basic Notations, Formulas, and Concepts 31.1 General Rules 3

1.1.1 Some notations 31.1.2 The rules to remove brackets 31.1.3 Short methods of multiplication 41.1.4 Fractions 51.1.5 The rules of handling with fractions 51.1.6 Fractional expressions 71.1.7 Proportions 71.1.8 Percentage 8

1.2 Decimal Fractions 81.2.1 Introduction 81.2.2 Handling with decimal fractions 81.2.3 Normalized form of numbers 101.2.4 Repeating decimals . . . . > 10

1.3 Rounding off Numbers. Approximate Numbers 111.3.1 Rounding off 11

- - 1.3.2 Approximate numbers 121.4 The Mathematical Induction Method 13

2 Sets. Real Numbers. Functions 152.1 Sets 15

2.1.1 Concept of a set 152.1.2 Subsets 162.1.3 Intervals 16

2.2 Real Numbers 172.2.1 Natural numbers 172.2.2 Rational numbers 192.2.3 Irrational numbers 192.2.4 Properties of arithmetic operations 19

2.3 Functions 202.3.1 Concept of a function 20

CONTENTS

2.3.2 Variables . , 202.3.3 Graph 21

2.4 Basic Characteristics of Functions 212.4.1 Monotonicity 212.4.2 Periodicity 232.4.3 Evenness and oddness 232.4.4 Boundedness 24

2.5 Inverse Functions 242.5.1 Definition 242.5.2 Inverse function for a monotonic one 252.5.3 Graph of the inverse function 25

2.6 Linear and Quadratic Functions. Modulus 252.6.1 Linear function 252.6.2 Quadratic function 262.6.3 Modulus 28

2.7 Degree Function 292.8 Exponential and Logarithmic Functions 31

2.8.1 Exponential function 312.8.2 Hyperbolic functions 312.8.3 Logarithmic function 32

Equations and Systems of Equations. Inequalities 353.1 General Concepts 35

3.1.1 Concept of the equation 353.1.2 Multiplicity of a root 363.1.3 Equivalent equations 363.1.4 Extraneous roots 37

3.2 Linear and Quadratic Equations 373.2.1 Linear equation 373.2.2 Quadratic equation 37

- 3.2.3 Biquadratic equation 393.3 Polynomials 39

3.3.1 Definitions 393.3.2 Horner's method 393.3.3 Polynomial algebra 403.3.4 Factorization of a polynomial 42

3.4 Algebraic Equations 433.4.1 Roots of an algebraic equation 433.4.2 On general formulas for the roots 433.4.3 Reduction of the degree of a polynomial 443.4.4 Binomial algebraic equation 443.4.5 Rational equations 45

3.5 Irrational and Modulus Equations 453.5.1 Irrational equations 453.5.2 Modulus equations 47

CONTENTS

3.6 Systems of Equations 483.6.1 Linear system, two equations 483.6.2 Linear system, three equations 493r6.3 ^Nonlinear system 51

3.7 Inequalities 523.7.1 Definitions 523.7.2 Basic properties of inequalities 533.7.3 Problems related with inequalities . 543.7.4 Domain of definition 553.7.5 Algebraic inequalities 553.7.6 Irrational inequalities 563.7.7 Transcendental inequalities 573.7.8 Inequalities with modulus symbol 58

4 Trigonometry 594.1 Trigonometric Functions 59

4.1.1 Trigonometric functions of an acute angle 594.1.2 Trigonometric functions of arbitrary values of ar-

gument 604.1.3 Properties of the trigonometric functions 62

4.2 Formulas of Trigonometry 644.2.1 Reduction formulas . 644.2.2 The basic formulas of trigonometry 65

4.3 Inverse Trigonometric Functions 664.3.1 Arcsine 664.3.2 Arccosine 674.3.3 Arctangent 674.3.4 Arccotangent ' 674.3.5 Formulas for inverse trigonometric functions . . . . 68

4.4 Trigonometric Equations and Inequalities 694.4.1 Simplest trigonometric equations 694.4.2 Equation reducible to the simplest one 694.4.3 Equation a sin x + b cos x = c 704.4.4 Trigonometric inequalities 71

5 Elements of Calculus 735.1 Sequences 73

5.1.1 Concept of a sequence . 735.1.2 Limit of a sequence 735.1.3 Monotonic and bounded sequences. Infinitesimal

sequences 755.1.4 Infinite limit 765.1.5 Arithmetic progression and series 765.1.6 Geometric progression and series 775.1.7 Infinitely decreasing geometric progression 78

CONTENTS

5.1.8 The number "e" - 805.2 Limit of Function 80

5.2.1 Definition and theorems 805.2.2 One-side limits , 825.2.3 Infinite limit 825.2.4 Limit at infinity 845.2.5 Rules to calculate the limits of functions 85

5.3 Continuity of Function. Discontinuities 885.3.1 Continuity 885.3.2 Discontinuities 89

5.4 Derivative. Differentiation Rules 915.4.1 Derivative 915.4.2 Derivatives of higher order 925.4.3 Application of derivatives 925.4.4 Inflection 935.4.5 Differentiation rules 94

5.5 Some Differential Equations 955.5.1 Introduction 955.5.2 Some simple differential equations 965.5.3 Some problems related to differential equations . . 97

5.6 Antiderivative. Indefinite Integral 985.6.1 Antiderivative 985.6.2 Indefinite integral 99

5.7 Definite Integral and its Applications 1015.7.1 Definition 1015.7.2 Properties of the definite integral 1035.7.3 Improper integrals: infinite interval 1045.7.4 Improper integrals: discontinuous function 1055.7.5 Some applications of the definite integral 106

5.8 Some Information about Series 1075.8.1 Convergence of a series 1075.8.2 Tests for convergence of series 1085.8.3 Series whose members are of arbitrary sign . . . . 1095.8.4 Power series 109

Combinatorics 1116.1 Permutations. Arrangements. Combinations I l l

6.1.1 Permutations I l l6.1.2 Factorial I l l6.1.3 Stirling formula 1126.1.4 Semi-factorial 1126.1.5 Arrangements 1126.1.6 Combinations 113

6.2 Binomial Formula and Applications 1136.2.1 Binomial formula 113

CONTENTS

6.2.2 Properties of the binomial coefficients 1146.2.3 Pascal triangle 1146.2.4 Sum of natural numbers in a certain power . . . . 115

7 Complex Numbers 1177.1 Basic Concepts 117

7.1.1 Notations 1177.1.2 Rules to operate with complex numbers 1177.1.3 Conjugate numbers 118

7.2 Algebraic Form of Complex Number 1187.3 Trigonometric and Exponential Forms 119

7.3.1 Vector interpretation 1197.3.2 Modulus and argument of a complex number . . . 1207.3.3 Trigonometric form 1207.3.4 Exponential form 1227.3.5 Euler formulas 1227.3.6 De Moivre formula 1237.3.7 Roots of complex numbers 124

7.4 Logarithms of Complex Numbers 1257.5 Complex Roots of Equations 125

8 Vectors. Coordinates. Symmetries 1278.1 Vectors. Projections 127

8.1.1 Vectors 1278.1.2 Projections 1308.1.3 Expansion in unit vectors 131

8.2 Scalar and Vector Products 1328.2.1 Scalar product ' 1328.2.2 Vector product 133

8.3 Coordinate Systems 1358.3.1 Coordinate axis 1358.3.2 Coordinate system in a plane 1358.3.3 Coordinate systems in space 1368.3.4 Polar coordinates 138

8.4 Displacement. Symmetry. Similarity 1398.4.1 Displacement ., 1398.4.2 Symmetry 1408.4.3 Spatial symmetry . 1428.4.4 Similarity 143

9 Geometry. Stereometry 1459.1 Points, Straight Lines and Angles in a Plane 145

9.1.1 Points and straight lines 1459.1.2 Segment 1469.1.3 Angle 146

CONTENTS

9.1.4 Degree 1479.1.5 Radian 1479.1.6 Intersection of straight lines 148

9.2 Triangles. Polygons 1499.2.1 Triangle 1499.2.2 Elements of a triangle 1509.2.3 Equal triangles 1519.2.4 Similar triangles 1539.2.5 Formulas for a triangle 1549.2.6 Polygons 1559.2.7 Parallelogram 1579.2.8 Rhombus 1589.2.9 Rectangle 1589.2.10 Square 1589.2.11 Trapezoid 159

9.3 Circle, Ellipse, Hyperbola, Parabola 1599.3.1 Circle 1599.3.2 Ellipse 1619.3.3 Hyperbola 1639.3.4 Parabola 1649.3.5 Curvature of a curve 164

9.4 Planes and Straight Lines in Space 1659.4.1 Parallelism of planes and lines 1659.4.2 Skew-lines 1659.4.3 Perpendicular 1669.4.4 Angle between two planes 1679.4.5 Dihedral angle 1689.4.6 Projection of a figure on a plane 168

9.5 Polyhedrons 169_.- 9.5.1 Polyhedral surface 169

9.5.2 Polyhedron 1699.5.3 Prism 1699.5.4 Parallelepiped 1709.5.5 Pyramid 1719.5.6 Frustum of a pyramid 1719.5.7 Regular polyhedrons 172

9.6 Bodies of Revolution 1739.6.1 Surface of revolution 1739.6.2 Cylinder 1739.6.3 Cone . . 1739.6.4 Sphere 1759.6.5 Spherical segment 1769.6.6 Spherical sector 1769.6.7 Spherical layer 177

CONTENTS

9.6.8 Torus " • • • • . 1779.7 Curvature of a Surface 178

10 Numerical Analysis 18110.1 Rounding off and Errors 18110.2 Approximation of Functions 183

10.2.1 Approximate formulas 18310.2.2 Function given by a table 18510.2.3 Method of least squares 188

10.3 Numerical Integration 18910.3.1 Trapezoid rule 19010.3.2 Formula of rectangles 19110.3.3 Simpson's formula 19110.3.4 Taylor approximation 19310.3.5 Improper integrals 193

10.4 Approximate Solution of Equations 19410.4.1 Bisection method 19410.4.2 Iteration method 19410.4.3 Newton's method 195

10.5 Approximate Solution of Differential Equations 19710.5.1 Euler's method 19710.5.2 Runge-Kutta method 198

11 Probability Theory 19911.1 Random Events and Probabilities 199

11.1.1 Random event 19911.1.2 Probability 20011.1.3 Bernoulli formula . . . .' 20211.1.4 Large numbers' law 203

11.2 Random Variables and Distributions 20411.2.1 Random variable 20411.2.2 Mean value and dispersion 205

11 PHYSICS 207

12 Physical Quantities and Systems of Units 20912.1 Basic Concepts. Laws of Physics 209

12.1.1 Physical quantities and measurements 20912.1.2 Equations in physics 21012.1.3 Physical-models 210

12.2 Systems of Units 21112.2.1 Base and derived units . . . 21112.2.2 Dimensions of physical quantities 21112.2.3 Standards of base units 213

CONTENTS

12.2 A Units of magnetic quantities in Gaussian system ofunits 214

12.2.5 Units of magnetic quantities in SI 21512.2.6 Relationship between SI and Gaussian units . . . . 217

12.3 The Method of Dimensional Analysis 21912.3.1 Dimensionless and dimensional units 21912.3.2 Example: velocity in free fall 21912.3.3 Example: flight range 22012.3.4 Example: viscous flow 22112.3.5 Example: speed of sound 22212.3.6 Example: velocity of waves 22312.3.7 Example: microscopic model of a real gas 22412.3.8 Example: time of relaxation in a gas 22412.3.9 Example: time of relaxation in plasma 22512.3.10Example: temperature dependence of black-body

radiation 226

13 Mechanics 22713.1 Kinematics 228

13.1.1 Kinematics of a particle 22813.1.2 Example: motion along an ellipse 22913.1.3 Velocity and acceleration 22913.1.4 Tangential and radial acceleration 23113.1.5 Rectilinear motion 23113.1.6 Circular uniform motion 23313.1.7 Kinematics of a rigid body 23313.1.8 Rotation about a fixed axis 23413.1.9 Plane motion of a solid 23413.l.lORotation about a fixed point 236

13.2 Dynamics 236• •-" 13.2.1 Basic concepts of classical dynamics 236

13.2.2 Momentum 23913.2.3 Determination of force on the basis of given motion 24013.2.4 Motion caused by given forces 24013.2.5 Restricted motion 242

13.3 Forces of Gravitation, Friction, and Elasticity 24313.3.1 The law of gravitation 24413.3.2 Friction and elasticity 245

13.4 Conservation Laws 24813.4.1 Conservation of momentum 24813.4.2 The center of mass 24913.4.3 The law of motion of the center of mass 25013.4.4 Jet propulsion 25113.4.5 Work and kinetic energy 25113.4.6 Potential energy 252

CONTENTS

13.4.7 Conservation of mechanical energy 25413.4.8 Collisions 255

13.5 Motion in a Central Gravitational Field 2571-3.5.1_ Kepler's laws of planetary motion 25713.5.2 Cosmic velocities 25813.5.3 Example: elliptic orbit of a satellite 259

13.6 Mechanical Equilibrium 26013.6.1 Conditions of equilibrium 26013.6.2 Plane system of forces 26113.6.3 Example: determination of forces of reaction . . . 26113.6.4 Statics and energy conservation 26313.6.5 The stability of equilibrium 263

13.7 Dynamics of a Solid 26413.7.1 The principal laws 26413.7.2 Moment of inertia 26513.7.3 Energy of rotation 26713.7.4 A gyroscope 267

13.8 Hydrostatics 26913.8.1 Pressure in a liquid 26913.8.2 Hydrostatic pressure 27013.8.3 Archimedes' principle and buoyant force 27113.8.4 Measurement of density 27113.8.5 Floating on the surface 272

13.9 Hydrodynamics 27313.9.1 Equation of continuity 27313.9.2 Bernoulli's principle 27413.9.3 Motion of a viscous fluid 27613.9.4 Turbulent flow of viscous fluid 279

14 Molecular Physics 281—14.1 Principles of Thermodynamics 282

14.1.1 Thermal equilibrium 28214.1.2 Parameters of the equilibrium state 28314.1.3 Equation of state for the ideal gas 28414.1.4 Gas thermometer 28614.1.5 Components and phases • 28614.1.6 Reversible and irreversible processes 28714.1.7 Internal energy 28714.1.8 Heat capacity 28814.1.9 Isoprocesses in the ideal Gas 28914.1.10 Efficiencyof a heat engine 29014.1.11 The second law of thermodynamics 29114.1.12 A refrigerator machine 29314.1.13Thermodynamic temperature 29314.1.14 Enthropy 294

CONTENTS

14.2 The Principles of Statistical Mechanics 29514.2.1 Thermal motion 29514.2.2 Molecular interaction 29614.2.3 Amount of substance 29714.2.4 Kinetic theory of an ideal gas 297

14.3 Statistical Distributions 29914.3.1 Distributions of different quantities 29914.3.2 Maxwell distribution 30014.3.3 Probabilities 30114.3.4 Calculation of mean values 30114.3.5 Boltzmann distribution 30214.3.6 Fluctuations 30314.3.7 Physical reasons for irreversibility 30414.3.8 Statistical meaning of enthropy 304

14.4 Real Gases 30514.4.1 Van der Vaals equation 30514.4.2 Experimental isotherms and phase transitions . . . 30614.4.3 Phase transitions 30714.4.4 Humidity of air 30814.4.5 Equilibrium of phases 309

14.5 Liquids 31014.5.1 Surface tension 31014.5.2 Capillary phenomena 311

14.6 Solids 31314.6.1 Crystals and amorphous bodies 31314.6.2 Elastic deformations 31314.6.3 Thermal expansion 316

14.7 Heat Exchange. Phase Transitions 31814.7.1 Thermal capacity 318

„ - 14.7.2 Heat of combustion 31914.7.3 Latent heat of phase transitions 319

15 Electricity and Magnetism 32115.1 Electrostatics : 321

15.1.1 Interaction of electric charges 32115.1.2 Electrostatic field 32215.1.3 Electrostatic field in dielectrics 32615.1.4 Electric field near conductors 32615.1.5 Capacitors 32715.1.6 Connection of capacitors 32815.1.7 Energy of electric field 329

15.2 Electric Current 33015.2.1 Ohm's law 33015.2.2 Series and parallel connection of resistors 33215.2.3 Measurements in direct current circuits 333

CONTENTS

15.2.4 Circuit with a source 33515.2.5 Kirchhoff's rules 33615.2.6 The work of electric current 33815.2.7 A power source in a circuit 33915.2.8 Faraday's laws of electrolysis 341

15.3 Magnetic Field 34115.3.1 Induction of magnetic field 34115.3.2 Ampere's force and Lorentz' force 34315.3.3 Magnetic field energy 34415.3.4 Magnetic field in substances 345

15.4 Electromagnetic Induction 34515.4.1 Faraday's law 34515.4.2 Inductance 346

15.5 Alternating Electric Current (AC) 34715.5.1 AC in circuits with one element 34715.5.2 Series HLC-circuit 34915.5.3 Parallel flLC-circuit 35015.5.4 Impedance of a circuit 35115.5.5 Resonance of voltages and resonance of currents . 35315.5.6 Power of alternating current 35315.5.7 Transformer 354

15.6 Electromagnetic Field 35515.6.1 Relative character of electric and magnetic fields . 35515.6.2 Invariants of electromagnetic field 35615.6.3 Maxwell's equations 356

16 Oscillations and Waves ( 35916.1 Classification of Oscillations 35916.2 Harmonic Oscillations 361

16.2.1 Kinematics of simple harmonic motion 36116.2.2 Vector diagrams for harmonic oscillations 362

16.3 Natural Oscillations of Simple Systems 36316.3.1 Differential equation of harmonic oscillator . . . . 36316.3.2 Initial conditions 36416.3.3 Transformations of energy in oscillations 36516.3.4 Nonlinear free oscillations 36616.3.5 Damped natural oscillations 36716.3.6 Damping by dry friction 370

16.4 Forced oscillations. Resonance 37016.4.1 Steady-state forced oscillations 37016.4.2 Resonance, curves of linear oscillator 37216.4.3 Resonance of velocity 37416.4.4 Energy in forced oscillations 37416.4.5 Transient processes 37516.4.6 Non-sinusoidal external force 375

CONTENTS

16.5 Parametric Resonance. Self-Excited Oscillations 37616.5.1 Parametric excitation of oscillations 37616.5.2 Self-excited oscillations 378

16.6 Oscillations of Complex Systems. Composition of Oscilla-tions 38016.6.1 Degenerate oscillatory systems 38016.6.2 Normal oscillations (modes) 38116.6.3 Coupled pendulums 38216.6.4 Forced oscillations of coupled pendulums 38416.6.5 Coupled electromagnetic circuits 38616.6.6 Standing waves as normal oscillations 386

16.7 Waves 38816.7.1 Waves of different physical nature 38816.7.2 Polarization of waves 38816.7.3 Kinematics of wave motion 38916.7.4 The speed of waves 39116.7.5 Energy transferred by waves 39316.7.6 Plane, spherical, and cylindrical waves 39416.7.7 Reflection and refraction of waves 39516.7.8 Interference of waves 39516.7.9 Standing waves 39716.7.10 Diffraction of waves 39816.7.11 Doppler effect 39816.7.12 Electromagnetic waves 40116.7.13Waveson the water 40416.7.14Speed of wave packets 406

17 Optics 40917.1 Geometrical Optics 409

17.1.1 The principal laws 40917.1.2 Plane mirrors 41317.1.3 Paraxial approximation and optical images . . . . 41417.1.4 Spherical mirrors 41517.1.5 Lenses 417

17.2 Optical Instruments 42017.2.1 Camera 42017.2.2 Diascope 42117.2.3 Magnifying glass 42217.2.4 Microscope 42317.2.5 Telescope 425

17.3 Interference of Light 42617.3.1 Interference and coherent light 42617.3.2 Interference fringes 42717.3.3 Young's double-slit experiment 42817.3.4 Localized interference patterns 430

CONTENTS

17.3.5 Multiple-ray interference 43317.3.6 The enlightenment of optical systems 434

17.4 Diffraction of Light 43417.4.1 The Huygens-Fresnel principle 43417.4.2 Diffraction spreading of a parallel light beam . . . 43417.4.3 Fresnel diffraction 43717.4.4 Fraunhofer diffraction 43817.4.5 Diffraction grating 43917.4.6 Dispersion spectrometer 44117.4.7 Holography 44217.4.8 Photometry 444

18 Relativistic and Quantum Physics 44718.1 The Theory of Relativity 44718.2 Relativistic Kinematics 448

18.2.1 Galilean transformation 44818.2.2 Insufficiency of classical concepts 44918.2.3 Main principles of the theory of relativity 45018.2.4 The relativity of simultaneity. Time dilation and

Lorentz contraction 45118.2.5 Lorentz transformations 45218.2.6 Relativistic interval 453

18.3 Relativistic Dynamics 45418.3.1 Relativistic momentum and energy 45418.3.2 Mass and energy 45518.3.3 Relativistic kinetic energy 45618.3.4 Relativistic transformation of energy and mo-

mentum 45718.3.5 Example: acceleration by a constant force 45718.3.6 Example: relativistic particle in magnetic field . . 45818.3.7 Transmutations of elementary particles 459

18.4 The Principles of Quantum Physics 46018.4.1 Uncertainty relations 46018.4.2 Wave-particle dualty 46218.4.3 Range of validity of classical theory 46318.4.4 Quanta of light—photons 46318.4.5 Photoelectric effect 46418.4.6 Light pressure 46518.4.7 Doppler effect 46518.4.8 Compton effect 465

18.5 The Structure of an Atom 46618.5.1 Bohr's model of the hydrogen atom 46618.5.2 Electron shells 46818.5.3 Light radiation of an atom 47018.5.4 Black-body radiation 470

CONTENTS

18.6 Atomic Nucleus 47118.6.1 Composition of atomic nuclei 47118.6.2 Radioactive decay 47418.6.3 Nuclear reactions 474

18.7 Elementary Particles 476

APPENDIX 479I Fundamental Physical Constants 479II Physical Quantities and their SI Units 480III Conversion of Gaussian Units into SI Units 483IV Conversion of Non-system Units into SI Units 484V Main Formulas of Electrodymamics in Gaussian Units and

in SI Units 485VI Atomic Elements and their Masses 487VII Table of Elementary Particles 488VIII Decimal Multiples to be Used with SI Units 489IX Relations between Fundamental Constants (in Gaussian

system of units) 489X Table of Mathematical Symbols 490

Index 491