Mathematical Handbook

The main purpose of the book Mathematical Handbook is to present school Students, applying to Mathematical Colleges and Universities, candidates for SAT I and SAT II, Students of Mathematics, Computer sciences, Economics, Engineering, Physics and other sciences full information on mathematics concepts, formulas, graphs, axioms, theorems and tables.

Moreover, this book is helpful to those Students who go to mathematics exams and competitions, where they can easily express their own ideas and thoughts by solving mathematical problems. Mathematical handbook is written according to the newest educational programs and textbooks of this subject all over the world.

We would like to thank of the reviewer Prof. Dr Sci Jordan Tabov and publisher - graduate engineer Stoyan Beev for their professional help and valuable advice.

Preface
Measures - transform factors
Mathematical symbols. Greek alphabet. Important constants
Discrete Mathematics

• Mathematical logic. Boolean operations
• Sets
• Systems of numeration
• Algorithms
• Combinatory
• Graph theory
• Coding theory

Algebra

• Arithmetic
• Number sets
• Set of natural numbers N
• Division of natural numbers
• Common fractions
• Decimal fractions. Percent and ratio
• Set of real numbers R
• Number intervals
• Absolute value (module) of a real number
• Powers and roots of real numbers
• Algebraic expressions - formulas for abbreviated multiplication and factorization of a polynomial to factors
• Logarithmic and antilogarithmic transformation of real numbers
• Number sequences
• Progressions
• Some numerical inequalities
• Set of complex numbers C
• Linear algebra
• Determinants
• Matrices
• Systems of linear equations
• Linear programming
• Linear programming in two dimensions geometric approach
• Simplex method
• Transportation problem
• Types of functions of one independent variable, equations, inequalities and simultaneous equations and inequalities
• Linear function, equation, inequalitie and simultaneous equations and inequalities
• Fractional-linear function, fractional-linear equations and inequalities
• Quadratic function, quadratic equations and inequalities
• Functions consisting of modules, module equations and inequalities
• Cubic function, cubic equations and inequalities
• Integer rational function of fourth power, equations and inequalities
• Fraction-rational function, fraction-rational equations and inequalities
• Power function, power equations and inequalities of integer degree
• Irrational function, irrational equations and inequalities
• Exponential function, exponential equations and inequalities
• Logarithmic function, logarithmic equations and inequalities
• Trigonometric functions, identities, equations and inequalities
• Inverse trigonometric functions and identities
• Hyperbolic functions and identities
• Inverse hyperbolic functions and identities
• Differential calculus of a function of one independent variable
• Properties of a numerical function
• Some non-elementary functions of one independent variable
• Some elementary functions of one independent variable - domains and ranges
• Limit of function
• Function continuity
• Derivative of a function
• Some higher order derivatives of functions
• Function analysis
• Differential calculus of two and more independent variables. Implicit functions
• Functions of two or more independent variables
• Partial derivatives
• Implicit functions
• Integral calculus
• Indefinite integrals
• Definite (Riemann’s) integrals
• Improper integrals
• Applications of definite integrals
• Integrals depending on a parameter
• Vector analysis
• Multiple integration
• Double integrals
• Triple integrals
• Application of double and triple integrals in geometry
• Curvilinear integrals
• Surface integrals
• Infinite series
• Number series
• Infinite series of functions
Complex analysis
• Operational calculus
• Laplace’s transforms
• Fourier’s transforms
• Differential equations
• Ordinary differential equations
• Partial differential equations
• Numerical analysis
• Numerical solution of equations
• Numerical differentiation
• Numerical integration
• Special functions and orthogonal polynomials
• Exponential integral functions
• Euler’s integrals
• Error function, Frenel integrals, Zeta function
• Bessel functions
• Hypergeometric functions
• Elliptic functions
• Orthogonal polynomials
• Probability theory
• Statistics
• Number characteristic of average
• Measuring the dispersion of data around a mean
• Number characteristic of a random variable
• Estimating the mean - confidence integral
• Test of hypotheses
• Linear regression and correlation
• Mathematics of finance
• Simple and compound interest
• Discount and annuity
• Capital public securities - bonds and stocks

Geometry

• Plane geometry
• Axioms of the plane geometry
• Angle. Proportional line segments
• Vectors
• Some loci in the plane
• Congruence. Criteria for congruence of triangles
• Homothety. Criteria for homothety of triangles
• Triangle
• Quadrilateral
• Polygon
• Circle
• Solid geometry
• Axioms of the solid geometry
• Parallel and perpendicularity
• Parallel and orthogonal projection. Angles in the space
• Geometric solid
• Edgy solids (polyhedrons)
• Circular solids
• Spherical trigonometry
• Areas of surfaces and volumes of edged and circular solids
• Area of a sphere, volume of a sphere and its segments
• Plane analytical geometry
• Coordinate systems in the plane
• Transformation of the coordinate system
• Vector - operations
• Division of a segment in a given ratio
• Equations of a straight line
• Conic sections
• General equation of second degree curves
• Some plane curves
• Solid analytical geometry
• Coordinate systems in the space
• Vector - operations
• Equations of a plane and a straight line
• Surfaces
• Differential geometry
• Applications