# Concise Handbook of Mathematics and Physics

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## Book Description

Concise Handbook of Mathematics and Physics presents a unified and coherent treatment of all the major aspects of modern elementary physics and mathematics. This complete text/reference includes definitions of fundamental notations and physical and mathematical quantities, formulas that express the laws of physics, axioms and theorems of mathematics, and more. The information is organized logically (instead of alphabetically) for better comprehension and quick, convenient access.

The book contains extensive cross-referencing between the mathematical and physical sections. reflecting the considerable overlap between these two areas of study and increasing the usefulness of this handbook. Fundamental concepts, theorems, and laws are demonstrated through numerous practical examples and tasks to help build problem-solving skills.

## 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

## Author(s)

### Biography

Alexander G. Alenitsyn (Hamburg, Germany) (Author) , Eugene I. Butikov (St. Petersburg State University, St. Petersburg, Russia) (Author) , Alexander S. Kondratyev (Central Washington University, Ellensburg, Washington, USA) (Author)