Concise Handbook of Mathematics and Physics

Table of Contents

MATHEMATICS
Basic Notations, Formulas, and Concepts
General Rules
Some Notations
The Rules to Remove Brackets
Short Methods of Multiplication
Fractions
The Rules of Handling with Fractions
Fractional Expressions
Proportions
Percentage
Decimal Fractions
Introduction
Handling with Decimal Fractions
Normalized Form of Numbers
Repeating Decimals
Rounding Off Numbers. Approximate Numbers
Rounding Off
Approximate Numbers
The Mathematical Induction Method
Sets. Real Numbers. Functions.
Sets
Concept of a Set
Subsets
Intervals
Real Numbers
Natural Numbers
Rational Numbers
Irrational Numbers
Properties of Arithmetic Operations
Functions
Concept of a Function
Variables
Graph
Basic Characteristics of Functions
Monotonicity
Periodicity
Evenness and Oddness
Boundedness
Inverse Functions
Definition
Inverse Function for a Monotonic One
Graph of the Inverse Function
Linear and Quadratic Functions. Modulus
Linear Function
Quadratic Function
Modulus
Degree Function
Exponential and Logarithmic Functions
Exponential Function
Hyperbolic Functions
Logarithmic Function
Equations and Systems of Equations. Inequalities
General Concepts
Concept of the Equation
Multiplicity of a Root
Equivalent Equations
Extraneous Roots
Linear and Quadratic Equations
Linear Equation
Quadratic Equation
Biquadratic Equation
Polynomials
Definitions
Horner's Method
Polynomial Algebra
Factorization of a Polynomial
Algebraic Equations
Roots of an Algebraic Equation
On General Formulas for the Roots
Reduction of the Degree of a Polynomial
Binomial Algebraic Equation
Rational Equations
Irrational and Modulus Equations
Irrational Equations
Modulus Equations
Systems of Equations
Linear System, Two Equations
Linear System, Three Equations
Nonlinear System
Inequalities
Definitions
Basic Properties of Inequalities
Problems Related with Inequalities
Domain of Definition
Algebraic Inequalities
Irrational Inequalities
Transcendental Inequalities
Inequalities with Modulus Symbol
Trigonometry
Trigonometric Functions
Trigonometric Functions of an Acute Angle
Trigonometric Functions of Arbitrary Values of Argument
Properties of the Trigonomic Functions
Formulas of Trigonometry
Reduction Formulas
The Basic Formulas of Trigonometry
Inverse Trigonomic Functions
Arcsine
Arccosine
Arctangent
Arccotangent
Formulas for Inverse Trigonomic Functions
Trigonometric Equations and Inequalities
Simplest Trigonomic Equations
Equation Reducible to the Simplest One
Equation a sin x + b cos x = c
Trigonometric Inequalities
Elements of Calculus
Sequences
Concept of a Sequence
Limit of a Sequence
Monotonic and Bounded Sequences. Infinitesimal Sequences
Infinite Limit
Arithmetic Progression and Series
Geometric Progression and Series
Infinitely Decreasing Geometric Progression
The Number "e"
Limit of Function
Definition and Theorems
One-Side Limits
Infinite Limit
Limit at Infinity
Rules to Calculate the Limits of Functions
Continuity of Function. Discontinuities
Continuity
Discontinuities
Derivative. Differentiation Rules
Derivative
Derivatives of Higher Order
Application of Derivatives
Inflection
Differentiation Rules
Some Differential Equations
Introduction
Some Simple Differential Equations
Some Problems Related to Differential Equations
Antiderivative. Indefinite Integral
Antiderivative
Indefinite Integral
Definite Integral and its Applications
Definition
Properties of the Definite Integral
Improper Integrals: Infinite Interval
Improper Integrals: Discontinuous Function
Some Applications of the Definite Integral
Some Information about Series
Convergence of a Series
Tests for Convergence of Series
Series Whose Members are of Arbitrary Sign
Power Series
Combinatorics
Permutations. Arrangements. Combinations
Permutations
Factorial
Stirling Formula
Semi-Factorial
Arrangements
Combinations
Binomial Formula and Applications
Binomial Formula
Properties of the Binomial Coefficients
Pascal Triangle
Sum of Natural Numbers in a Certain Power
Complex Numbers
Basic Concepts
Notations
Rules to Operate with Complex Numbers
Conjugate Numbers
Algebraic Form of Complex Number
Trigonometric and Exponential Forms
Vector Interpretation
Modulus and Argument of a Complex Number
Trigonometric Form
Exponential Form
Euler Formulas
De Moivre Formula
Roots of Complex Numbers
Logarithms of Complex Numbers
Complex Roots of Equations
Vectors. Coordinates. Symmetries
Vectors. Projections
Vectors
Projections
Expansion in Unit Vectors
Scalar and Vector Products
Scalar Product
Vector Product
Coordinate Systems
Coordinate Axis
Coordinate System in a Plane
Coordinate Systems in Space
Polar Coordinates
Displacement. Symmetry. Similarity
Displacement
Symmetry
Spatial Symmetry
Similarity
Geometry. Stereometry
Points, Straight Lines, and Angles in a Plane
Points and Straight Lines
Segment
Angle
Degree
Radian
Intersection of Straight Lines
Triangles. Polygons
Triangle
Elements of a Triangle
Equal Triangles
Similar Triangles
Formulas for a Triangle
Polygons
Parallelogram
Rhombus
Rectangle
Square
Trapezoid
Circle, Ellipse, Hyperbola, Parabola
Circle
Ellipse
Hyperbola
Parabola
Curvature of a Curve
Planes and Straight Lines in Space
Parallelism of Planes and Lines
Skew-Lines
Perpendicular
Angle Between Two Planes
Dihedral Angle
Projection of a Figure on a Plane
Polyhedrons
Polyhedral Surface
Polyhedron
Prism
Parallelepiped
Pyramid
Frustum of a Pyramid
Regular Polyhedrons
Bodies of Revolution
Surface of Revolution
Cylinder
Cone
Sphere
Spherical Segment
Spherical Sector
Spherical Layer
Torus
Curvature of a Surface
Numerical Analysis
Rounding Off and Errors
Approximation of Functions
Approximate Formulas
Function Given by a Table
Method of Least Squares
Numerical Integration
Trapezoid Rule
Formula of Rectangles
Simpson's Formula
Taylor Approximation
Improper Integrals
Approximate Solution of Equations
Bisection Method
Iteration Method
Newton's Method
Approximate Solution of Differential Equations
Euler's Method
Runge-Kutta Method
Probability Theory
Random Events and Their Probabilities
Random Event
Probability
Bernoulli Formula
Large Numbers' Law
Random Variables and Distribution
Random Variable
Mean Value and Dispersion
PHYSICS
Physical Quantities and Systems of Units
Basic Concepts. Law of Physics
Physical Quantities and Measurements
Equations in Physics
Physical Models
Systems of Units
Based and Derived Units
Dimensions of Physical Quantities
Standards of Base Units
Units of Magnetic Quantities in Gaussian System of Units
Units of Magnetic Quantities in SI
Relationship between SI and Gaussian Units
The Method of Dimensional Analysis
Dimensionless and Dimensional Units
Example: Velocity in Free Fall
Example: Flight Range
Example: Viscous Flow
Example: Speed of Sound
Example: Velocity of Waves
Example: Microscopic Model of a Real Gas
Example: Time of Relaxation in a Gas
Example: Time of Relaxation in Plasma
Example: Temperature Dependence of Black-Body Radiation
Mechanics
Kinematics
Kinematics of a Particle
Example: Motion Along an Ellipse
Velocity and Acceleration
Tangential and Radial Acceleration
Rectilinear Motion
Circular Uniform Motion
Kinematics of a Rigid Body
Rotation About a Fixed Axis
Plane Motion of a Solid
Rotation About a Fixed Point
Dynamics
Basic Concepts of Classical Dynamics
Momentum
Determination of Force on the Basis of Given Motion
Motion Caused by Given Forces
Restricted Motion
Forces of Gravitation, Friction, and Elasticity
The Law of Gravitation
Friction and Elasticity
Conservation Laws
Conservation of Momentum
The Center of Mass
The Law of Motion of the Center of Mass
Jet Propulsion
Work and Kinetic Energy
Potential Energy
Conservation of Mechanical Energy
Collisions
Motion in a Central Gravitational Field
Kepler's Laws of Planetary Motion
Cosmic Velocities
Example: Elliptic Orbit of a Satellite
Mechanical Equilibrium
Conditions of Equilibrium
Plane System of Forces
Example: Determination of Forces of Reaction
Statics and Energy Conservation
The Stability of Equilibrium
Dynamics of a Solid
The Principal Laws
Moment of Inertia
Energy of Rotation
A Gyroscope
Hydrostatics
Pressure in a Liquid
Hydrostatic Pressure
Archimedes' Principle and Buoyant Force
Measurement of Density
Floating on the Surface
Hydrodynamics
Equation of Continuity
Bernoulli's Principle
Motion of a Viscous Fluid
Turbulent Flow of Viscous Fluid
Molecular Physics and Thermodynamics
Principles of Thermodynamics
Thermal Equilibrium
Parameters of the Equilibrium State
Equation of State for the Ideal Gas
Gas Thermometer
Components and Phases
Reversible and Irreversible Phases
Internal Energy
Heat Capacity
Isoprocesses in the Ideal Gas
Efficiency of a Heat Engine
The Second Law of Thermodynamics
A Refrigerator Machine
Thermodynamic Temperature
Enthropy
The Principles of Statistical Mechanics
Thermal Motion
Molecular Interaction
Amount of Substance
Kinetic Theory of an Ideal Gas
Statistical Distributions
Distributions of Different Quantities
Maxwell Distribution
Probabilities
Calculation of Mean Values
Boltzmann Distribution
Fluctuations
Physical Reasons of Irreversibility
Statistical Meaning of Enthropy
Real Gases
Van der Vaals Equation
Experimental Isotherms and Phase Transitions
Phase Transitions
Humidity of Air
Equilibrium of Phases
Liquids
Surface Tension
Capillary Phenomena
Solids
Crystals and Amorphous Bodies
Elastic Deformations
Thermal Expansion
Heat Exchange. Phase Transitions
Thermal Capacity
Heat of Combustion
Latent Heat of Phase Transitions
Electricity and Magnetism
Electrostatics
Interaction of Electric Charges
Electrostatic Field
Electrostatic Field in Dielectrics
Electric Field Near Conductors
Capacitors
Connection of Capacitors
Energy of Electric Field
Electric Current
Ohm's Law
Series and Parallel Connection of Resistors
Measurements in Direct Current Circuits
Circuit with a Source
Kirchhoff's Rules
The Work of Electric Current
A Power Source in a Circuit
Faraday's Laws of Electrolysis
Magnetic Field
Induction of Magnetic Field
Ampere's Force and Lorentz' Force
Magnetic Field Energy
Magnetic Field in Substances
Electromagnetic Induction
Faraday's Law
Inductance
Alternating Electric Current (AC)
AC in Circuits with One Element
Series RLC-Circuit
Parallel RLC-Circuit
Impedance of a Circuit
Resonance of Voltages and Resonance of Currents
Power of Alternating Current
Transformer
Electromagnetic Field
Relative Character of Electric and Magnetic Fields
Invariants of Electromagnetic Field
Maxwell's Equations
Oscillations and Waves
Classification of Oscillations
Harmonic Oscillations
Kinematics of Simple Harmonic Motion
Vector Diagrams for Harmonic Oscillations
Natural Oscillations of Simple Systems
Differential Equation of Harmonic Oscillator
Initial Conditions
Transformations of Energy in Oscillations
Nonlinear Free Oscillations
Damped Natural Oscillations
Damping by Dry Friction
Forced Oscillations. Resonance
Steady-State Forced Oscillations
Resonance Curves of Linear Oscillator
Resonance of Velocity
Transformations of Energy in Forced Oscillations
Transient Processes
Non-Sinusoidal External Force
Parametric Resonance. Self-Excited Oscillations
Parametric Excitation of Oscillations
Self-Excited Oscillations
Oscillations of Complex Systems. Composition of Oscillations
Degenerate Oscillatory Systems
Normal Oscillations (Modes)
Coupled Pendulums
Forced Oscillations of Coupled Pendulums
Coupled Electromagnetic Circuits
Standing Waves as Normal Oscillations
Waves
Waves of Different Physical Nature
Polarization of Waves
Kinematics of Wave Motion
The Speed of Waves
Energy Transferred by Waves
Plane, Spherical, and Cylindrical Waves
Reflection and Refraction of Waves
Interference of Waves
Standing Waves
Diffraction of Waves
Doppler Effect
Electromagnetic Waves
Waves on the Water
Speed of Wave Packets
Optics
Geometrical Optics
The Principal Laws
Plane Mirrors
Paraxial Approximation and Optical Images
Spherical Mirrors
Lenses
Optical Instruments
Camera
Diascope
Magnifying Glass
Microscope
Telescope
Interference of Light
Interference and Coherent Light
Interference Fringes
Young's Double-Slit Experiment
Localized Interference Patterns
Multiple-Ray Interference
The Enlightenment of Optical Systems
Diffraction of Light
The Huygens-Fresnel Principle
Diffraction Spreading of a Parallel Light Beam
Fresnel Diffraction
Fraunhofer Diffraction
Diffraction Grating
Dispersion Spectrometer
Holography
Photometry
Relativistic and Quantum Physics
The Theory of Relativity
Relativistic Kinematics
Galilean Transformation
Insufficiency of Classical Concepts
Main Principles of the Theory of Relativity
The Relativity of Simultaneity. Time Dilation and Lorentz Contraction
Lorentz Transformations
Relativistic Interval
Relativistic Dynamics
Relativistic Momentum and Energy
Mass and Energy
Relativistic Kinetic Energy
Relativistic Transformation of Energy and Momentum
Example: Acceleration by a Constant Force
Example: Relativistic Particle in Magnetic Field
Transmutations of Elementary Particles
The Principles of Quantum Physics
Uncertainty Relations
Wave-Particle Duality
Range of Validity of Classical Theory
Quanta of Light-Photons
Photoelectric Effect
Light Pressure
Doppler Effect
Compton Effect
The Structure of an Atom
Bohr's Model of the Hydrogen Atom
Electron Shells
Light Radiation of an Atom
Black-Body Radiation
Atomic Nucleus
Composition of Atomic Nuclei
Radioactive Decay
Nuclear Reactions
Elementary Particles
APPENDIX
Fundamental Physical Constants
Physical Quantities and Their SI Units
Conversion of Gaussian Units into SI Units
Conversion of Non-System Units into SI Units
Main Formulas of Electrodynamics in Gaussian Units and in SI Units
Atomic Elements and Their Masses
Table of Elementary Particles
Decimal Multiples to be Used with SI Units
Relations between Fundamental Constants (in Gaussian System of Units)
Table of Mathematical Symbols
Index