Plant of low-voltage and high-voltage equipment. Low-Voltage and High-Voltage Equipment Plant III

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The previous, 12th edition (1980) came out with a radical revision made by a large team of authors from the GDR, edited by G. Grosche and W. Ziegler. Numerous corrections have been made to this edition. For students, engineers, scientists, teachers.

1.1.3.3. Table indefinite integrals.

General instructions. 1. The constant of integration is omitted everywhere except when the integral can be represented in various forms with various arbitrary constants.

Editorial
1. TABLES AND GRAPHS
1.1. TABLES
1.1.1 Tables of elementary functions
1. Some common constants A1) 2. Squares, cubes, roots A2). 3. Powers of integers from 1 to 100 B9). 4. Reciprocals of C1). 5. Factorials and their reciprocals C2). 6 Some powers of numbers 2, 3 and 5 C3). 7. Decimal logarithms C3). 8. Antilogarithms C6) 9. Natural values ​​of trigonometric functions C8) 10. Exponential, hyperbolic and trigonometric functions (for x from 0 to 1.6) D6). 11. Exponential functions (for x from 1.6 to 10.0) D9). 12. Natural logarithms E1). 13. Circumference E3). 14. Area of ​​a circle E5). 15. Elements of a circle segment E7). 16. Converting a degree measure to a radian F1). 17. Proportional parts F1). 18. Table for quadratic interpolation F3)
1 1.2. Special Function Tables
1. Gamma function F4). 2 Bessel (cylindrical) functions F5). 3. Legendre polynomials (spherical functions) F7). 4. Elliptic integrals F7). 5 Poisson distribution F9). 6 Normal distribution G1). 7. X2-distribution G4). 8. /-student distribution G6). 9. z-distribution G7). 10. F-distribution (distribution v2) G8). 11. Critical numbers for the Wilcoxon test (84). 12. X-distribution of Kolmogorov-Smirnov (85).
1.1.3. Integrals and sums of series
1 Table of sums of some numerical series (86). 2. Table of expansion of elementary functions into power series (87). 3 Table of indefinite integrals (91). 4 Table of some definite integrals (PO).
1.2. GRAPHS OF ELEMENTARY FUNCTIONS
1.2.1 Algebraic functions FROM
1 Entire rational functions A13). 2. Fractional rational functions A14). 3. Irrational functions A16).
1.2.2. Transcendent Functions
1. Trigonometric and inverse trigonometric functions A17). 2. Exponential and logarithmic functions A19) 3. Hyperbolic functions A21).
1.3. KEY CURVES
1.3.1. Algebraic curves
1 3rd order curves A23). 2. 4th order curves A24).
1 3.2. Cycloids
1.3.3. Spirals
1.3.4. Chain line and tractrix
2. ELEMENTARY MATHEMATICS
2.1. ELEMENTARY APPROXIMATE CALCULATIONS
2.1.1. General information
1. Representation of numbers in positional number system A30). 2. Errors and rules for rounding numbers A31)
2.2. COMBINATORICS
2 2 1 Basic combinatorial functions 1 Factorial and gamma function A34) 2 Binomial coefficients A34). 3 Polynomial factor A35)
2 2 2. Binomial and polynomial formulas 1 Newton's binomial formula A35) 2 Polynomial formula A35)
2 2.3 Statement of problems of combinatorics
2 24 Substitutions
1. Substitutions A36). 2. The group of permutations to elements A36). 3. Fixed Point Substitutions A36). 4 Permutations with a given number of cycles A37) 5 Permutations with repetitions A37)
2 2 5. Placements 137 1 Placements A37) 2 Placements with repetitions A37). 2 2 6 Combinations 1 Combinations A38). 2 Combinations with repetitions A38).
2.3. FINITE SEQUENCES, SUMS, PRODUCTS, AVERAGES
2 3 1 Notation of sums and products
2 3.2 End sequences 1 Arithmetic progression A39) ^2 Geometric progression A39)
2 3 3 Some finite sums
2 3 4 Average values
2.4. ALGEBRA
2 4 1. General concepts 1 Algebraic expressions A40) 2 Meanings of algebraic expressions A40) 3 Polynomials A41) 4 Irrational expressions A41). 5 Inequalities A42) 6. Elements of group theory A43)
2 4.2 Algebraic equations 1 Equations A43) 2 Equivalent transformations A44) 3 Algebraic equations A45) 4. General theorems A48). 5 System algebraic equations A50)
24 3 Transcendental equations
2.4 4 Linear algebra 1. Vector spaces A51) 2. Matrices and determinants A56). 3. Siams linear equations A61) 4 Linear transformations A64). 5 Eigenvalues ​​and eigenvectors A66)
2.5. ELEMENTARY FUNCTIONS
2 5 1. Algebraic functions 1 Entire rational functions A69) 2 Fractional rational functions A70) 3 Irrational algebraic functions A74)
2 52 Transcendent functions 1. Trigonometric functions and their inverses A74). 2 Exponential and logarithmic functions A79). 3 Hyperbolic functions and their inverses A80).
2.6. GEOMETRY
2 6 1. Planimefia
26 2 Stereometry 1 Straight lines and planes in space A85) 2 Dihedral, polyhedral and solid angles A86) 3 Polyhedra A86) 4 Bodies formed by moving lines A88)
2.6.3. Rectilinear trigonometry 1. Solving triangles A90) 2. Application in elementary geodesy A91)
2 6 4. Spherical trigonometry
1. Geometry on the sphere A92). 2. Spherical triangle A92) 3 Solution of spherical triangles A92).
2.6.5. Coordinate systems
1. Coordinate systems on the plane A95). 2 Coordinate systems in space A97)
2.6.6. Analytic geometry
1. Analytic geometry in the plane A99) 2 Analytic geometry in space B04)
3. FUNDAMENTALS OF MATHEMATICAL ANALYSIS
3.1. DIFFERENTIAL AND INTEGRAL CALCULUS OF FUNCTIONS OF ONE AND SEVERAL VARIABLES
3.1.1. Real numbers
1. The system of axioms of real numbers B10) 2. Natural, integer and rational numbers B11) 3 The absolute value of a number B12). 4. Elementary inequalities B12)
3.1.2. Point sets in R"
3.1 3. Sequences
1. Number sequences B14) 2 Point sequences B15)
3.1.4. Real Variable Functions
1. Function of one real variable B16) 2 Functions of several variable variables B23).
3.1 5. Differentiation of functions of one real variable
1. Definition and geometric interpretation of the first derivative Examples B25) 2 Higher order wires B26).
3. Properties of differentiable functions B27) 4 Monotonicity and convexity of functions B28).
5. Extrema and inflection points B29) 6 Elementary study of the function B30).
3.1.6. Differentiation of functions of several variables. N 2M
1. Partial derivatives, geometric interpretation B30) 2. Total directional differential, gradient B31) 3. Theorems on differentiable functions of several variables B32)
4. Differentiable mapping of the space Rn into Rm, functional definitions i el u. implicit functions; existence theorems B33) 5 Change of variables in differential expressions B35). 6. Extrema of functions of several variables B36)
3.1 7. Integral calculus of functions of one variable
1. Definite integrals B38) 2 Properties of definite integrals B39) 3 Indefinite integrals B39). 4. Properties of indefinite integrals B41) 5 Integration of rational functions B42)
6. Integration of other classes of functions B44) 7 Improper integrals B47) 8 Geometric and physical applications of definite integrals. B51)
3.1.8. Curvilinear integrals
1. Curvilinear integrals of the 1st kind (integrals over the length of a curve) B53) 2 Realization and calculation of curvilinear integrals of the 1st kind B53) general view) B54) 4. Properties and calculation of curvilinear integrals of the 2nd kind B54).
5. Independence of the curvilinear integrals oi of the integration path B56) 6. Geometical and physical applications of the curvilinear integrals B57)
3.1.9. Integrals depending on a parameter
1. Definition of integral depending on parameter B57) 2 Properties of integrals depending on oi parameter B57). 3. Improper integrals depending on parameter B58) 4 Examples of integrals depending on parameter B60)
3.1.10. Double integrals 2b0
1. Definition of a double integral and elementary properties B60) 2 Calculation of double integrals B61).
3. Change of variables in double integrals B62) 4 Geometrical and physical applications of double integrals B63)
3.1.11. Triple Integrals
1. Definition of the triple integral and elementary properties B63) 2 Calculation of multiple hhicirals B64). 3. Change of Variables in Triple Integrals B65). 4 Geometrical and physical applications of triple integrals B65).
3.2. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
3.2.1. Calculus of variations
1. Statement of the problem, examples and basic concepts B87). 2. Euler-Lagrange theory B88). 3. The theory of Hamilton - Jacobi B94). 4. Inverse problem of the calculus of variations B95). 5. Numerical Methods B95).
3.2.2. Optimal control
1. Basic concepts B98) 2. Pontryagin's maximum principle B98). 3. Discrete systems C03) 4. Numerical methods C04).
3.3. DIFFERENTIAL EQUATIONS
3.3.1. Ordinary differential equations
1 General concepts. Existence and uniqueness theorems C05) 2. First order differential equations C06). 3. Linear differential equations and linear systems C13). 4. General non-linear differential equations C25). 5. Stability C25) 6. Operator method for solving ordinary differential equations C26) 7. Boundary value problems and eigenvalue problems C27).
3.3.2. Partial Differential Equations
1. Basic concepts and special methods solutions C31) 2. Partial differential equations of the 1st order C33). 3. Partial differential equations of the 2nd order C39).
3.4. COMPLEX NUMBERS. FUNCTIONS OF A COMPLEX VARIABLE
3.4.1. General remarks
3.4 2. Complex numbers. Riemann sphere. Areas
1. Definition of complex numbers Field of complex numbers C57). 2. Conjugate complex numbers Modulus of a complex number C58). 3. Geometric interpretation of C58). 4. Trigonometric and exponential forms of complex numbers C58). 5 Degrees, roots C59). 6. Riemann sphere. Jordan curves. Regions C59).
3 4.3. Functions of a complex variable
3.4.4. The most important elementary functions
1. Rational functions C61) 2 Exponential and logarithmic functions C61) 3 Trigonometric and hyperbolic functions C64).
3.4.5. Analytic Functions i. Derivative C65) 2 Cauchy-Riemann differentiability conditions C65) 3 Analytic functions C65).
3.4.6. Curvilinear integrals in the complex domain
1. Integral of a function of a complex variable C66). 2. Independence of the path of integration C66).
3. Indefinite integrals C66) 4 Basic formula of integral calculus C66). 5. Cauchy integral formulas C66)
3.4.7. Expansion of analytic functions in a series
1. Sequences and series C67). 2 Functional rows. Power series C68). 3. Taylor series C69). 4 Laurent series C69). 5. Classification singular points C69). 6. Behavior of analytic functions at infinity C70).
3.4.8. Deductions and their application
1. Residues C70). 2. Residue theorem C70). 3. Application to the calculation of definite integrals C71).
3 49 Analytic continuation 1 Principle of analytic continuation C71). 2 Symmetry principle (Schwarz) C71)
3 4.10 Inverse functions Riemann surfaces
1 Univalent functions, inverse functions C72) 2. Riemann surface of the function z = |/w C72). 3. Riemann surface of the function z - Ln w C73).
3 4 11 Conformal mappings
1 The concept of a conformal mapping C73) 2. Some simple conformal mappings C74).
4. ADDITIONAL CHAPTERS
4.1. SETS, RELATIONS, MAPPINGS
4 1 1 Basic concepts of mathematical logic
1 Algebra of logic (propositional algebra, propositional logic) C76) 2 Predicates C79)
4 1 2. Basic concepts of set theory
1. Sets, elements C80). 2 Subsets of C80)
4 1 3 Operations on sets
1 Union and intersection of sets C81). 2. Difference, symmetric difference, complement of sets C81) 3 Euler-Venn diagrams C81) 4. Cartesian product of sets C82) 5. Generalized union and intersection C82)
4.1.4 Relations and mappings
1. Relations C82) 2 Equivalence relation C83) 3 Order relation C83). 4. Mappings C84).
5. Sequences and families of sets C85) 6 Operations and algebras C85).
4.1 5 Cardinality of sets
1. Equivalence C86). 2 Countable and uncountable sets C86)
4.2. VECTOR CALCULUS
4 2 1 Vector algebra
1 Basic concepts C86). 2. Scalar multiplication and addition C86). 3. Multiplication of vectors C88).
4 Geometric Applications of Vector Algebra C89).
4 2 2. Vector analysis
1 Vector functions of scalar argument C90) 2. Fields (scalar and vector) C91). 3. Scalar field gradient C93). 4. Curvilinear integral and potential in a vector field C94). 5 Surface integrals in vector fields C95). 6. Divergence of a vector field C97). 7. Vector field curl C98).
8. Laplace operator and vector field gradient C99). 9. Calculation of complex expressions (Hamilton operator) C99). 10. Integral formulas D00) 11 Definition of a vector field by its sources and vortices D01) 12. Dyads (tensors of rank II) D02)
4.3. DIFFERENTIAL GEOMETRY
4 3.1 Flat curves
1 Ways to specify plane curves. Plane curve equation D05). 2 Local elements of a plane curve D06) 3 Points of a special type D07). 4 Asymptotes D09) 5 Evolute and involute D10). 6 Envelope of a family of curves D10).
4 3 2 Spatial curves
1 Ways of specifying curves in space D10). 2 Local elements of a curve in space D10)
3 Main theorem of the theory of curves D11).
4.3.3. surfaces
1. Methods for defining surfaces D12) 2 Tangent plane and normal to the surface D12).
3. Metric properties of surfaces D13). 4 Surface curvature properties D14). 5. Main theorem of the theory of surfaces D16). 6 Geodesic lines on the surface D17).
4.4. FOURIER SERIES, FOURIER INTEGRALS, AND THE LAPLACE TRANSFORM
4 4.1. Fourier series
1 General concepts D18). 2. Table of some Fourier expansions D19) 3 Numerical harmonic analysis D23).
4 4 2. Fourier integrals
1 General concepts D25). 2 Table of Fourier transforms D26).
4.4 3 Laplace transform
1 General concepts D37) 2 Application of the Laplace transform to the solution of ordinary differential equations with initial conditions D38) 3 Table of the inverse Laplace transform of fractional rational functions D38)
5. PROBABILITY THEORY AND MATHEMATICAL STATISTICS
5.1. PROBABILITY THEORY
5 1 1 Random events and their probabilities
1 Random events D41) 2 Axioms of the theory of probability D42). 3 The classic definition of faith! event probability D43) 4 Conditional probabilities D43) 5. Total probability Bayes formula D43)
5 1 2 Random variables
1 Discrete random variables D44) 2 Continuous random variables D45)
5 1 3 Moments of distribution
1 Discrete case D46) 2 Continuous case D47)
5 1 4 Jurassic random ages (multivariate random variables)
1 Discrete random vectors D48) 2 Continuous random vectors D49) 3 Boundary distributions D49) 4 Moments of a multidimensional random variable D49) 5. Conditional distributions D50)
6 Independentib random variables D50) 7 Regression dependence D50) 8 Functions oi of random variables D51)
5 1 5 Characteristic functions
1 Properties of characteristic functions D52). 2 Inversion formula and uniqueness theorem D52) 3 Limit theorem for characteristic functions D52) 4 Generating functions D53)
5 Characteristic functions of multidimensional random variables D53).
5 1 6 Limit theorems
1 law big numbers D53) 2 Moivre-Laplace limit theorem D54) 3 Central limit theorem D54)
5.2. MATH STATISTICS
5 2 1 Samples
1 Histogram and empirical distribution function D55). 2 Sample function D56) 3 Some important distributions D57)
5 2 2 Parameter estimation
1 Properties of point estimates D57) 2 Methods for obtaining estimates D58). 3 Confidence estimates D59)
5 2 3 Hypothesis testing (tests)
1 Statement of the problem D60) 2 General theory D60) 3 r-test D61) 4 /-test D61) 5 Wilcoxon test D61). 6 X-criterion D62) 7. Case of additional parameters D63) 8 Kolmogorov-Smirnov agreement criterion D63)
5 2 4 Correlation and regression
1 Estimation of correlation and pei ression characteristics by samples D64) 2 Checking innoiejbi р = 0
in the case of a normally distributed 1general population D64)
6. MATHEMATICAL PROGRAMMING
6.1. LINEAR PROGRAMMING,6 11 Statement of the problem of linear programming and the simplex method
1 General setting of giving, i eoms! logical interpretation and solution for sch with noisy variables D66)
2 Canonical view of the LLP, image of the vertex in the simplex table D68) 3 Simplex method with a given initial table D69) 4 Obtaining the initial vertex D71). 5 Degenerate case and its treatment using the simplex method D73) 6 Duality in linear programming D73).
7 Modified methods, additional change to task D75)
6.2. TRANSPORT CHALLENGE
6 2 1 Linear transport problem
62 2 Omitting the initial solution
62 3 Transport method
6.3. TYPICAL LINEAR PROGRAMMING APPLICATIONS
6.3.1 Capacity utilization
6.3.2. Mixture problem
6.3.3. Distribution, planning, comparison
6.3.4. Cutting, shift planning, coating
6.4. PARAMETRIC LINEAR PROGRAMMING
6.4 1 Problem statement
6 4.2. Solution Method for the Case of a One-Parameter Objective Function
6.5. INTEGER LINEAR PROGRAMMING
6 5 1. Statement of the problem, geometric interpretation
6.5.2. Gomory Section Method
1. Purely integer linear programming problems D87). 2. Mixed integer linear programming problems D88).
6.5.3 Branch method
6.5 4 Comparison of methods
7. ELEMENTS OF NUMERICAL METHODS AND THEIR APPLICATIONS
7.1. ELEMENTS OF NUMERICAL METHODS
7.1.1. Errors and their accounting
7.1.2. Computational methods
1. Solution of linear systems of equations D91). 2. Linear eigenvalue problems (D95).
3. Nonlinear Equations D96) 4. Systems of Nonlinear Equations D98) 5 Approximation D99) 6 Interpolation E02) 7 Approximate Calculation of Integrals E06) 8 Approximate Differentiation E10). 9 Differential Equations E10).
7 1.3 Implementation of the numerical model in electronic computers
I. Criteria for the choice of method E16). 2. Control methods E16). 3. Calculation of functions E17).
7.1 4 Nomography and slide rule
1 Relations between two variables - functional scales E18) 2. Slide rule E19). 3. Nomograms of points on straight lines and grid nomograms E19).
7.1 5 Handling empirical numerical material
1. Method of least squares E21). 2. Other alignment methods E22).
7.2. COMPUTER ENGINEERING
7.2.1. Electronic computers (computers)
1. Introductory remarks E23) 2. Representation of information and computer memory E23) 3 Exchange channels E24). 4 Program E24). 5. Programming E24). 6. Computer control E26). 7. Mathematical (software) E26). 8. Performing work on a computer E26)
7.2.2 Analog computers
1. The principle of the analog device computer science E27). 2 Computing elements of an analog computer E27). 3. Programming Principle for Solving Systems of Ordinary Differential Equations (E29). 4 Quality programming E30)
Bibliography
Subject index

I. N. Bronstein and K. A. Semendyaev’s handbook on mathematics for engineers and students of higher educational institutions has firmly gained popularity not only in our country, but also abroad. The eleventh edition was published in 1967. Further edition of the reference book was suspended, as it no longer met modern requirements.

Decimal logarithms.
Explanations for tables of logarithms and antilogarithms. Table 1.1.1.7 is used to find the decimal logarithms of numbers. First, for a given number, the characteristic ei about the logarithm is found, and then the mantissa from the table. For three-digit numbers, the mantissa is located at the intersection of the line at the beginning of which (column N) are the first two digits of the given number, and the column corresponding to the third digit of our number. If the given number has more than three significant digits, linear interpolation must be applied. In this case, the interpolation correction is found only on the fourth significant digit of the number; it makes sense to make a correction for the fifth digit only when the first significant digit of the given number is 1 or 2.

To find a number by its decimal logarithm, use table 1.1.1.8 (table of antilogarithms) *). The argument in this table is the mantissa of the given logarithm. At the intersection of the row, which is determined by the first two digits of the mantissa (column m), and the column corresponding to the third digit of the mantissa, the digital composition of the desired number is found in the antilogarithm table. An interpolation correction must be applied to the fourth digit of the mantissa. The characteristic of the logarithm allows you to put a comma in the result.

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• Handbook of mathematics for engineers and students of higher educational institutions, Bronstein I.N., Semendyaev K.A., 1986
• Non-standard methods for solving equations and inequalities, Reference book, Olehnik S.N., Potapov M.K., Pasichenko P.I., 1991
• Mathematics, School reference book, grades 7-11, Definitions, formulas, schemes, theorems, algorithms, Chernyak A.A., Chernyak Zh.A., 2018

The following tutorials and books.

I. N. BRONSHTEIN K. A. SEMENDYAEV
MATHEMATICS HANDBOOK FOR ENGINEERS AND STUDENTS
22.11B 88
UDC 51
Authors from the GDR who participated in the revision of the edition:
DIPL.-MATH. P. BECKMANN, DR. M. BELGER, DR. H. BENKER,
D.R. M. DEWEB, PROF. D.R. H. ERFURTH, DIPL.-MATH. H. GENTEMANN,
D.R. P. GOTHNER, DOZ. D.R. S. GOTTWALD, DOZ. D.R. G. GROSCHE,
DOZ. D.R. H. HILBIG, DOZ. D.R. R. HOFMANN, NPT H. KASTNER,
D.R. W. PURKERT, DR. J. VOM SCHEIDT, DIPL.-MATH. TH. VETTERMANN, D.R. v. WfjNSCH, PROF. D.R. E. ZEIDLER.
A Handbook of Mathematics for Engineers P university students.
Bronstein I. N., Semendyaev K. A.-M.: Science.
Main edition financial and mathematical literature, 1981.

Teubner Publishing House, GDR, 1979 ) Publishing house "Science",Main editionphysical and mathematical Literature, 1980

CONTENT
Editorial
1. TABLES AND GRAPHS
1.1. TABLES
1.1.1. Tables of elementary functions
1. Some common constants (12). 2. Squares, cubes, corn (12). 3. Degrees of integers from 1 to 100 (30). 4. Reciprocals (32). 5. Factorials and their reciprocals (34). 6. Some powers of numbers 2, 3 and 5 (35). 7. Decimal logarithms (36). 8. Antilogarithms (38) 9. Natural values ​​of trigonometric functions (40). 10. Exponential, hyperbolic and trigonometric functions (48). 11. Exponential functions (for x from 1.6 to 10.0) (51). 12. Natural logarithms (S3). 13. Circumference (56). 14. Area of ​​a circle (58). 15. Circle segment elements (60). 16. Converting a degree measure to a radian (64). 17. Proportional parts (65). 18. Table for quadratic interpolation (67).

1.1.2. Special Function Tables
1. Gamma function (68). 2. Bessel (cylindrical) functions (69). 3. Legendre polynomials (spherical functions) (71). 4. Elliptic integrals (72). 5. Poisson distribution (74). 6. Normal distribution (75). 7. chi distribution (78). 8. Student's r-distribution (80). 9. z-distribution (81). 10. F-distribution (distribution u3) (82). 11. Critical numbers for the Wilcoxon test (88). 12. Kolmogorov - Smirnov distribution (89).

1.1.3. Integrals and sums of series
1. Table of sums of some numerical series (90). 2. Table of expansion of some functions into power series (92). 3. Table of indefinite integrals (95). 4. Table of some definite integrals (122).

1.2. GRAPHS OF ELEMENTARY FUNCTIONS
1.2.1. Algebraic functions
1. Entire rational functions (126). 2. Fractional-rational functions (127). 3. Irrational functions (130).
1.2.2 Transcendent functions
1. Trigonometric and inverse trigonometric functions (131). 2. Exponential and logarithmic functions (133). 3. Hyperbolic functions (136).

1.3. KEY CURVES
1.3.1. Algebraic curves
1. Curves of the 3rd order (138). 2 Curves of the 4th order (139).
1.3.2. Cycloids
1.3.3. Spirals
1.3.4. Chain line and tractrix

2. ELEMENTARY MATHEMATICS 2.1. ELEMENTARY APPROXIMATE CALCULATIONS
2.1.1. General information
1. Representation of numbers in positional number system (147). 2. Errors and rules for rounding numbers (148).
2.1.2. Elementary Error Theory
1. Absolute and relative errors (149). 2. Approximate error limits for the function (149). 3. Approximate formulas (149).
2.1.3. Elementary Approximate Graphical Method
1. Finding the zeros of the function (150). 2. Graphical differentiation (150). 3. Graphical integration (151).

2.2. COMBINATORICS
2.2.1. Basic combinatorial functions
1. Factorial and gamma function (151). 2. Binomial coefficients (152). 3. Polynomial coefficient (153).
2.2.2. Binomial and polynomial formulas
1. Newton's binomial formula (153). 2. Polynomial formula (154).
2.2.3. Statement of problems of combinatorics
2.2.4. Permutations
1. Permutations (154). 2. The permutation group of k elements (155). 3. Permutations with a fixed point (156). 4. Permutations with a given number of cycles (156). 5. Permutations with repetitions (156).
2.2.5. Accommodations
1. Placements (157). 2. Placements with repetitions (157).
2.2.6. Combinations
1. Combinations (157). 2. Combinations with repetitions (158).

2.3. FINITE SEQUENCES, SUMS, PRODUCTS, AVERAGES
2.3.1. Notation of sums and products
2.3.2. End sequences
1. Arithmetic progression (159). 2. Geometric progression (159).
2.3.3. Some final sums
2.3.4. Averages

2.4. ALGEBRA
2.4.1. General concepts
1. Algebraic expressions (161). 2. Values ​​of algebraic expressions (161). 3. Polynomials (162). 4. Irrational expressions (163). 5. Inequalities (163). 6. Elements of group theory (165).
2.4.2. Algebraic equations
1. Equations (165). 2. Equivalent transformations (166). 3. Algebraic equations (167). 4. General theorems (171). 5. System of algebraic equations (173).
2.4.3. Transcendental Equations
2.4.4. Linear algebra
1. Vector spaces (175). 2. Matrices and determinants (182). 3. Systems of linear equations (189). 4. Linear transformations (192). 5. Eigenvalues ​​and eigenvectors (195).

2.5. ELEMENTARY FUNCTIONS
2.5.1. Algebraic functions
1. Entire rational functions (199). 2. Fractional-rational functions (201). 3. Irrational algebraic functions (205).
2.5.2. Transcendent Functions
1. Trigonometric functions and their inverses (206). 2. Exponential and logarithmic functions (212). 3. Hyperbolic functions and their inverses (213).

2.6. GEOMETRY
2.6.1. Planimetry
2.6.2. Stereometry
1. Straight lines and planes in space (220). 2. Dihedral, polyhedral and solid angles (220). 3. Polyhedra (221). 4. Bodies formed by moving lines (223).
2.6.3. Rectilinear Trigonometry
1. Solution of triangles (225). 2. Application in elementary geodesy (227).
2.6.4. Spherical trigonometry
1. Geometry on the sphere (228). 2. Spherical triangle (228). 3. Solution of spherical triangles (229).
2.6.5. Coordinate systems
1. Coordinate systems on the plane (232). 2. Coordinate systems in space (234).
2.6.6. Analytic geometry
1. Analytic geometry on the plane (237). 2. Analytic geometry in space (244).

3. FUNDAMENTALS OF MATHEMATICAL ANALYSIS
3.1. DIFFERENTIAL AND INTEGRAL CALCULUS OF FUNCTIONS OF ONE AND SEVERAL VARIABLES
3.1.1. Real numbers
1. System of axioms of real numbers (252). 2. Natural, integer and rational numbers (253). 3. The absolute value of the number (254). 4. Elementary inequalities (254).
3.1.2. Point sets in R"
3.1.3. Sequences
1. Numerical sequences (257). 2. Sequences of points (259).
3.1.4. Real Variable Functions
1. Function of one real variable (260). 2. Functions of several real variables (269).
3.1.5. Differentiation of functions of one real variable
1. Definition and geometric interpretation of the first derivative. Examples (272). 2. Derivatives of higher orders (273). 3. Properties of differentiable functions (275). 4. Monotonicity and convexity of functions (277). 5. Extreme points and inflection points (278). 6. Elementary investigation of a function (279).
3.1.6. Differentiation of functions of several variables
1. Partial derivatives, geometric interpretation (280). 2. Total differential, directional derivative, gradient (280). 3. Theorems on differentiable functions of several variables (282). 4. Differentiable mapping of the space R" into R"1; functional determinants; implicit functions; existence theorems for a solution (284). 5. Change of variables in differential expressions (286). 6. Extrema of functions of several variables (288).
3.1.7. Integral calculus of functions of one variable
1. Definite integrals (291). 2. Properties of definite integrals (292). 3. Indefinite integrals (293). 4. Properties of indefinite integrals (295). 5. Integration of rational functions (297). 6. Integration of other classes of functions (300). 7. Improper integrals (30S). 8. Geometric and physical applications of definite integrals (312).
3.1.8. Curvilinear integrals
1. Curvilinear integrals of the 1st kind (integrals over the length of a curve) (3I5). 2. Existence and calculation of curvilinear integrals of the first kind (315). 3. Curvilinear integrals of the second kind (projection integrals and general integrals) (316). 4. Properties and calculation of curvilinear integrals of the second kind (316). 5. Independence of curvilinear integrals from the path of integration (318). 6. Geometric and physical applications of curvilinear integrals (320).
3.1.9. Integrals depending on a parameter
1. Definition of an integral depending on the parameter (321). 2. Properties of integrals depending on a parameter (321). 3. Improper integrals depending on a parameter (322). 4. Examples of integrals depending on the parameter (324).
3.1.10. Double integrals
1. Definition of the double integral and elementary properties (326). 2. Calculation of double integrals (327). 3. Change of variables in double integrals (328). 4. Geometric and physical applications of double integrals (328).
3.1.11. Triple Integrals
I. Definition of the triple integral and the simplest properties (330). 2. Calculation of triple integrals (330). 3. Change of variables in triple integrals (331). 4. Geometric and physical applications of triple integrals (332).
3.1.12. Surface integrals
1. The area of ​​a smooth surface (333). 2. Surface integrals of the 1st and 2nd kind (334). 3. Geometric and physical applications of the surface integral (337).
3.1.13. Integral formulas
1. Formula of Ostrogradsky - Gauss. Green's formula (336). 2. Green's formulas (339). 3. Formula. Stokes (339). 4. Improper curvilinear, double, surface and triple integrals (339). 5. Multidimensional integrals depending on a parameter (341).
3.1.14. Endless rows
1. Basic concepts (343). 2. Criteria for the convergence or divergence of series with non-negative terms (344). 3. Series with arbitrary members. Absolute convergence (347). 4. Functional sequences. Functional series (349). Power series (352). 6. Analytic functions. Taylor series. Expansion of elementary functions in a power series (357).
3.1.15. Endless works

3.2. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
3.1.1. Calculus of variations
1. Statement of the problem, examples and basic concepts (365). 2. Euler-Lagrange theory (366). 3. The theory of Hamilton - Jacobi (376). 4. Inverse problem of the calculus of variations (377). 5. Numerical methods (378).
3.22. Optimal control
1. Basic concepts (381). 2. Pontryagin's maximum principle (383). 3. Discrete systems (390). 4. Numerical methods (391).

3.3. DIFFERENTIAL EQUATIONS
3.3.1. Ordinary differential equations
1. General concepts. Existence and uniqueness theorems (393). 2. Differential equations of the 1st order (395). 3. Linear differential equations and linear systems 404 4. General non-linear differential equations (420). 5. Stability 421 6. Operator method for solving ordinary differential equations (422). 7. Boundary value problems and eigenvalue problems (424).
3.3.2. Partial Differential Equations
1. Basic concepts and special methods of solution (428). 2. Equations in partial derivatives of the 1st order (431). 3. Equations in partial derivatives of the 2nd order (440).

3.4. COMPLEX NUMBERS. FUNCTIONS OF A COMPLEX VARIABLE
3.4.1. General remarks
3.4.2. Complex numbers. Riemann sphere. Areas
1. Definition of complex numbers. Field of complex numbers (466). 2. Conjugate complex numbers. Complex number modulus (467). 3. Geometric interpretation 468 4. Trigonometric and exponential forms of complex numbers (468). 5. Degrees, roots (469). 6. Riemann sphere. Jordan curves. Regions (470).
1.4.3. Functions of a complex variable
1.4.4. The most important elementary functions
1. Rational functions (473). 2. Exponential and logarithmic functions (474). 3. Trigonometric and hyperbolic functions 475
3.4.5. Analytic Functions
1. Derivative (476). 2. Cauchy-Riemann differentiability conditions (476). 3. Analytic functions 476
3.4.6. Curvilinear integrals in the complex domain
1. Integral of a function of a complex variable (477). 2. Independence from the path of integration (478). 3. Indefinite integrals (478). 4. Basic formula of the integral calculus (478). 5. Cauchy integral formulas 478
3.4.7. Expansion of analytic functions in a series
1. Sequences and series (479). 2. Functional rows. Power series (480). 3. Taylor series (481). 4. Laurent series (481). 5. Classification of singular points (482). 6. Behavior of analytic functions at infinity (482).
3.4.8. Deductions and their application
1. Deductions (483). 2. Residue theorem (483). 3. Application to the calculation of definite integrals (484).
3.4.9. Analytic continuation
1. The principle of analytic continuation (484). 2. Principle of symmetry (Schwartz) (485).
3.4.10. Inverse functions. Riemann surfaces
1. Univalent functions, inverse functions (485). 2. Riemann surface of a function (486). 3. Riemann surface of the function r=Lnw (486).
3.4.11. Conformal mapping
1. The concept of a conformal mapping (487). 2. Some simple conformal mappings (488).

4. ADDITIONAL CHAPTERS
4.1. SETS, RELATIONS, MAPPINGS
4.1.1. Basic concepts of mathematical logic
1. Algebra of logic (algebra of propositions, logic of propositions) (490). 2. Predicates (494).
4.1.2 Basic concepts of set theory
1. Sets, elements (496). 2. Subsets (496).
4.1.3. Operations on sets
1. Union and intersection of sets (496). 2. Difference, symmetric difference, complement of sets (496). 3. Euler - Venn diagrams (497). 4. Cartesian product of sets (497). 5. Generalized union and intersection 498
4.1.4. Relationships and mappings
1. Relations (498). 2. Equivalence relation (499). 3. Order relation (500). 4. Mappings (501). 5. Sequences and families of sets (502). 6. Operations and algebras 502
4.1.5. Power of sets
1. Equivalence (503). 2. Countable and uncountable sets 503

4.2. VECTOR CALCULUS 4.2.1. Vector algebra
1. Basic concepts (5.03). 2. Multiplication by a scalar and addition (504). 3. Multiplication of vectors (505). 4. Geometric applications of vector algebra (507).
4.2.2. Vector Analysis
1. Vector functions of a scalar argument (508). 2. Fields (scalar and vector) 510 3. Gradient of a scalar field 513 4. Curvilinear integral and potential in a vector field 515 5. Surface integrals in vector fields 6. Divergence of a vector field 519 7. Vector field rotor (520). 8. The Laplace Operator and the Gradient of a Vector Field (521) 9. Calculation of complex expressions (Hamilton operator) (522). 10. Integral formulas 523 11. Determination of a vector field from its sources and vortices 525 12. Dyads (tensors of rank II) (526).

4.3. DIFFERENTIAL GEOMETRY
4.3.1. Flat curves
1. Methods for setting plane curves. Plane curve equation (531). 2 Local elements of a plane curve (532). 3. Points of a special type (534). 4. Asymptotes (536). 5. Evolute and involute (537). 6. Envelope of a family of curves 538
4.3.2. Spatial curves
1. Ways of specifying curves in space (538). 2. Local elements of a curve in space 538 3. Main theorem of the theory of curves (540).
4.3.3. surfaces
1. Methods for defining surfaces (540). 2 Tangent plane and surface normal (541). 3. Metric properties of surfaces (543). 4. Surface curvature properties 545 5. Main theorem of the theory of surfaces (547). 6. Geodesic lines on the surface 548

4.4. FOURIER SERIES, FOURIER INTEGRALS, AND THE LAPLACE TRANSFORM
4.4.1. Fourier series
1. General concepts (549). 2. Table of some expansions in the Fourier series (551). 3. Numerical harmonic analysis 556
4.4.2. Fourier integrals
I. General concepts (559). 2. Tables of Fourier transforms (561).
4.4.3. Laplace transform
1. General concepts (571). 2. Application of the Laplace transform to the solution of ordinary differential equations with initial conditions (573). 3. Table of the inverse Laplace transform of fractional rational functions (574).

5. PROBABILITY THEORY AND MATHEMATICAL STATISTICS
5.1. PROBABILITY THEORY
5.1.1. Random events and their probabilities
1. Random events (577). 2. Axioms of the theory of probability (578). 3. The classical definition of the probability of an event (579). 4. Conditional probabilities 580 5. Full probability. Bayes formula (580).
5.1.2. random variables
I. Discrete Random Variables 581 2. Continuous random variables 583
5.1.3. Moments of distribution
I. Discrete case 585 2. Continuous case 587
5.1 4 Random vectors (multidimensional random variables)
1. Discrete random vectors 588 2. Continuous random vectors 588 3. Boundary distributions 589 4. Moments of a multidimensional random variable 589 5. Conditional distributions. 6. Independence of random variables 590 7. Regression dependence (591). 8. Functions of random variables 592
5.1.5. Characteristic functions
1. Properties of characteristic functions 593 2. The inversion formula and the uniqueness theorem (594). 3. Limit theorem for characteristic functions (594). 4. Generating functions 595 5. Characteristic functions of multidimensional random variables 595
5.1.6. Limit theorems
1. Laws of large numbers (595). 2. Limit theorem of De Moivre - Laplace (596). 3. Central limit theorem (597).

5.2. MATH STATISTICS
5.2.1. Samples
1. Histogram and empirical distribution function (598). 2. Sample functions (600). 3. Some important distributions (600).
5.2.2. Parameter Estimation
1. Properties of point estimates (601). 2. Methods for obtaining estimates (602). 3. Confidence estimates (604).
5.2.3. Hypothesis testing (tests)
1. Statement of the problem (606). 2. General theory 606 3. meriterium (607). 4. F-test (607), 5. Wilcoxon test (607). 6. Chi test (608). 7. The case of additional parameters (609). 8. Kolmogorov-Smirnov agreement criterion (610).
5.24. Correlation and regression
1. Evaluation of correlation and regression characteristics for samples (611). 2. Testing the hypothesis p = 0 in the case of a normally distributed general population (612). 3. General problem of regression (612).

6. MATHEMATICAL PROGRAMMING
6.1. LINEAR PROGRAMMING
1. General formulation of the problem, geometric interpretation and solution of problems with two variables (613). 2. Canonical view, image of the vertex in the simplex table (615). 3. Simplex method for given 7. Modified methods, additional changes to the problem (625).

6.2. TRANSPORT CHALLENGE
6.2.1. Linear transport problem
6.2.2. Finding the Initial Solution
6.23. transport method

6.3. TYPICAL LINEAR PROGRAMMING APPLICATIONS
6.3.3. Distribution, planning, comparison
6.3.4. Cutting, shift planning, coating

6.4. PARAMETRIC LINEAR PROGRAMMING
6.4.1. Formulation of the problem
6.4.2. Solution Method for the Case of a One-Parameter Objective Function

6.5. INTEGER LINEAR PROGRAMMING 6.5.1. Problem statement, geometric interpretation
6.5.2 Gomory cut method
6.5.3. Branch method
6.5.4. Comparison of methods

7. ELEMENTS OF NUMERICAL METHODS AND THEIR APPLICATIONS
7.1. ELEMENTS OF NUMERICAL METHODS
7.1.1. Errors and their accounting
7.1.2. Computational methods
1. Solution of linear systems of equations (649). 2. Linear eigenvalue problems 653 3. Nonlinear equations (655). 4. Systems of nonlinear equations 657 5. Approximation 659 6. Interpolation (663). 7. Approximate calculation of integrals (668). 8. Approximate differentiation 673 9. Differential Equations 674
7.1.3. Implementation of the Numerical Model in Electronic Computers
1. Criteria for choosing a method (681). 2. Management methods (682). 3. Calculation of functions (682).
7.1.4. Nomography and slide rule
1. Relations between two variables - functional scales (685). 2. Logarithmic (counting) ruler (686). 3. Nomograms of points on straight lines and grid nomograms (687).
7.1.5. Handling Empirical Numerical Material
1. Method of least squares (688). 2. Other methods of alignment (690).

7.2. COMPUTER ENGINEERING
7.2.1. Electronic computers (computers)
1. Introductory remarks (691). 2. Representation of information and computer memory (692). 3. Exchange channels (693). 4. Program (693). 5. Programming (694). 6. Computer control (695). 7. Mathematical (software) software (696). 8. Performing work on a computer (696).
7.2.2. Analog computers
1. The principle of the design of analog computing technology (697). 2. Computing elements of an analog computer (697). 3. Principle of programming in solving systems of ordinary differential equations (699). 4. Quality programming (700).

Literature
Universal designations
Subject index

EDITORIAL
Handbook of I. N. Bronstein and K. A. Semendyaev in mathematics for engineersand students of technical universities has firmly gained popularity not only in our country, butand abroad. The eleventh edition was published in 1967. Further publication of the reference book was suspended, as it no longer met modern requirements.The revision of the handbook was carried out at the initiative of the publishing house "Teubner», with the consent of the authors, a large team of specialists in the GDR (where previously referencedNick withstood 16 editions). A mutual decision was made to release this revisedtanny version co-published:in the GDR - the publishing house "Teubner" - in German;in the USSR - the main edition of the physical and mathematical literature of the publishing house"Science" - in Russian.As a result of the revision, the guide was not only enriched with new informationon those sections of mathematics that were presented earlier, but was supplementedand new sections: calculus of variations and optimal control, mathematical logic and set theory, computational mathematics and basicinformation on computing.At the same time, the general methodological style of the handbook was preserved, allowingand get factual help on finding formulas or tabular data, and familiarize yourself with the basic concepts (or restore them to memory); For a better understanding of the concepts, a large number of examples are given.In connection with such a thorough revision of the handbook, the entire text was rewrittentranslated from German.During the preparation of the Russian edition, some revision was made in order toto take into account, if possible, the requirements of the programs of domestic universities. This pererabotka is mainly associated with a change in the designations and terminology that we haveand in the GDR are not identical. Some sections for the Russian edition have been rewrittenagain - these are the first sections of the chapters on algebra, mathematical logic,set theory. The sections devoted to complex variables, the calculus of variations, and optimal control have undergone a less significant alteration.computational mathematics.To reduce the size of the handbook compared to originally plannedoption omitted some sections that are necessary for a narrower circle specialists. Some sections of the handbook were left without revision due tothe very short time allotted for the preparation of this publication. For example, in thisThe edition omits the section on tensor calculus. In this regard, section"Differential Geometry" should be rewritten in somewhat more detail andchange the presentation. The Computational Mathematics section says a lotabout computational methods and little is given to computational mathematics proper.In the section "Calculation of Variations and Optimal Control" there is not enough attentionniya is given to optimal control. However it takes a long time to complete this workand, most importantly, reader feedback. Therefore, the editorialwith a request to all who will use the guide to send their commentsand suggestions for improving the handbook so that they can be taken into account in furtherthe most work on it.Please send your proposals to the address: 117071, Moscow, Leninsky Prospect, 15, Main editorial office of physical and mathematical literature of the Nauka publishing house, editorialmathematical reference books.

Download the book Bronstein I. N., Semendyaev K. A. Handbook of mathematics. For engineers and university students. Publishing house "Science", Moscow, 1981

Brief excerpt from the beginning of the book(machine recognition)

I.N. BRONSHTEIN
K.A.SEMENDYAEV
DIRECTORY
on
MATH
FOR ENGINEERS AND STUDENTS
THIRTEENTH EDITION, REVISED
MOSCOW "NAUKA"
MAIN EDITION
PHYSICAL AND MATHEMATICAL LITERATURE
1986
SmmeebyUo
BBC 22.11
B68
UDC 51
Authors from the GDR who took part in the preparation of the guide:
P. BECKMANN, M. BELGER, H. BENKER, M. DEWEB,
H. ERFURTH, H. GENTEMANN, S. GOTTWALD, P. GUTHNER,
G. GROSCHE, H. HILBIG, R. HOFMANN, H. KASTNER,
W. PURKERT, J. von SCHEIDT, TH. VETTERMANN,
V. WUNSCH, E. ZEIDLER
Bronstein I. N., Semendyaev K. A. Handbook of mathematics
for engineers and students of technical colleges. - 13th ed., corrected. - M.: Nauka,
Ch. ed. Phys.-Math. lit., 1986.- 544 p.
The previous, 12th edition A980) came out with a radical revision,
produced by a large team of authors from the GDR, edited by
G. Grosche and W. Ziegler. This edition includes numerous
fixes.
For students, engineers, scientists, teachers.
Ilya Nikolaevich Bronstein
Konstantin Adolfovich Semendyaev
HANDBOOK FOR MATH
for engineers and university students
Editor A. I. Stern
Art editor T. N. Kolchenko
Technical editors V. N. Kondakova, S. Ya. Shklnr
Proofreaders T S Weisberg, L S Somova
I B 12490
Handed over to the set 08/27/85. Signed for printing 27.05.86 Format
70 x 100/16. Book and magazine paper for offset printing.
Timestamp headset. Offset printing. Conv. p l. 44.2 Uel cr-ott 88.4.
Uch.-ed. l 72.22. Circulation 250,000 copies. Order 60. Price 4 rubles. 10 k.
Order of the Red Banner of Labor, Nauka publishing house
Main edition of physical and mathematical literature
117071 Moscow V-71, Leninsky prospect, 15
Order of the October Revolution, Order of the Red Labor
Znamya Leningrad Production and Technical Association
"Printing Yard" named after A. M. Gorky Soyuzpoligrafprom at
USSR State Committee for Publishing and Printing
and book trade
197136, Leningrad, P-136, Chkalovsky pr., 15.
1702000000 - 106
[email protected])-86
4
© Teubner Publishing House,
GDR, 1979
© Publishing house "Science",
Main edition
physical and mathematical
Literature, 1980,
with changes, 1986
CONTENT
Edition 10
1. TABLES AND GRAPHS
1.1. TABLES
1.1.1 Tables of elementary functions 11
1. Some common constants A1) 2. Squares, cubes, roots A2). 3. Powers of integers
numbers from 1 to 100 B9). 4. Reciprocals of C1). 5. Factorials and their reciprocals C2).
6 Some powers of numbers 2, 3 and 5 C3). 7. Decimal logarithms C3). 8. Antilogarithms C6) 9.
Natural values ​​of trigonometric functions C8) 10. Exponential, hyperbolic and trigonometric
functions (for x from 0 to 1.6) D6). 11. Exponential functions (for x from 1.6 to 10.0) D9). 12.
Natural logarithms E1). 13. Circumference E3). 14. Area of ​​a circle E5). 15. Circle Segment Elements
E7). 16. Converting a degree measure to a radian F1). 17. Proportional parts F1). 18. Table for
quadratic interpolation F3)
1 1.2. Special Function Tables 64
1. Gamma function F4). 2 Bessel (cylindrical) functions F5). 3. Legendre polynomials (spherical
functions) F7). 4. Elliptic integrals F7). 5 Poisson distribution F9). 6 Normal distribution
G1). 7. X2-distribution G4). 8. /-student distribution G6). 9. z-distribution G7). 10. F-distribution
(distribution v2) G8). 11. Critical numbers for the Wilcoxon test (84). 12. X-distribution
Kolmogorov-Smirnov (85).
1.1.3. Integrals and sums of series 86
1 Table of sums of some numerical series (86). 2. Table of expansion of elementary functions into power functions
rows (87). 3 Table of indefinite integrals (91). 4 Table of some specific
integrals (PO).
1.2. GRAPHS OF ELEMENTARY FUNCTIONS
1.2.1 Algebraic functions FROM
1 Entire rational functions A13). 2. Fractional rational functions A14). 3. Irrational
functions A16).
1.2.2. Transcendent functions 117
1. Trigonometric and inverse trigonometric functions A17). 2. Exponential and logarithmic
functions A19) 3. Hyperbolic functions A21).
1.3. KEY CURVES
1.3.1. Algebraic curves 123
1 3rd order curves A23). 2. 4th order curves A24).
1 3.2. Cycloids 125
1.3.3. Spirals 128
1.3.4. Chain line and tractor 129
2. ELEMENTARY MATHEMATICS
2.1. ELEMENTARY APPROXIMATE CALCULATIONS
2.1.1. General information 130
1. Representation of numbers in positional number system A30). 2. Errors and rounding rules
numbers A31)
1*
CONTENT
2 1 2 Elemental error theory 131
1 Absolute and relative errors A31) 2. Approximate error limits of the function A32)
3 Approximate formulas A32)
2 1.3. Elementary Approximate Graphical Methods. 1. Finding the zeros of the function /(x) A32). 2 Graphic
differentiation A33) 3 Graphical integration A33)
2.2. COMBINATORICS
2 2 1 Basic combinatorial functions 134
1 Factorial and gamma function A34) 2 Binomial coefficients A34). 3 Polynomial
factor A35)
2 2 2. Binomial and polynomial formulas 135
1 Newton binomial formula A35) 2 Polynomial formula A35)
2 2.3 Statement of problems of combinatorics 135
2 24 Substitutions 136
1. Substitutions A36). 2. The group of permutations to elements A36). 3. Fixed point substitutions
A36). 4 Permutations with a given number of cycles A37) 5 Permutations with repetitions A37)
2 2 5. Accommodations 137
1 Placements A37) 2 Placements with repetitions A37).
2 2 6 Combinations 138
1 Combinations A38). 2 Combinations with repetitions A38).
2.3. FINAL SEQUENCES, SUMS,
PRODUCTS, AVERAGES
2 3 1 Notation of sums and products 138
2 3.2 End sequences 138
1 Arithmetic progression A39) ^2 Geometric progression A39)
2 3 3 Some finite sums 139
2 3 4 Averages 139
2.4. ALGEBRA
2 4 1. General concepts 140
1 Algebraic expressions A40) 2 Algebraic expression values ​​A40) 3 Polynomials A41)
4 Irrational expressions A41). 5 Inequalities A42) 6. Elements of group theory A43)
2 4.2 Algebraic equations 143
1 Equations A43) 2 Equivalent transformations A44) 3 Algebraic equations A45) 4. General
Theorem A48). 5 System of algebraic equations A50)
24 3 Transcendental equations 150
2.4 4 Linear Algebra 151
1. Vector spaces A51) 2. Matrices and determinants A56). 3. Systems of linear equations A61)
4 Linear transformations A64). 5 Eigenvalues ​​and eigenvectors A66)
2.5. ELEMENTARY FUNCTIONS
2 5 1. Algebraic functions 169
1 Entire rational functions A69) 2 Fractional rational functions A70) 3 Irrational
algebraic functions A74)
2 52 Transcendent functions 174
1. Trigonometric functions and their inverses A74). 2 Exponential and logarithmic functions
A79). 3 Hyperbolic functions and their inverses A80).
2.6. GEOMETRY
2 6 1. Planimefia 183
26 2 Stereometry 185
1 Lines and planes in space A85) 2 Dihedral, polyhedral and solid angles A86) 3
Polyhedra A86) 4 Bodies formed by moving lines A88)
CONTENT
2.6.3. Rectilinear Trigonometry 189
1. Solving triangles A90) 2. Application in elementary geodesy A91)
2 6 4. Spherical trigonometry 192
1. Geometry on the sphere A92). 2. Spherical triangle A92) 3 Solution of spherical triangles
A92).
2.6.5. Coordinate systems 194
1. Coordinate systems on the plane A95). 2 Coordinate systems in space A97)
2.6.6. Analytic geometry 199
1. Analytic geometry in the plane A99) 2 Analytic geometry in space B04)
3. FUNDAMENTALS OF MATHEMATICAL ANALYSIS
3.1. DIFFERENTIAL AND INTEGRAL CALCULUS.
FUNCTIONS OF SINGLE AND MULTIVARIABLES
3.1.1. Real numbers 210
1. System of axioms of real numbers B10) 2. Natural, integer and rational numbers B11) 3 Abeolkn-
value of number B12). 4. Elementary inequalities B12)
3.1.2. Point sets in R" 212
3.1 3. Sequences 214
1. Number sequences B14) 2 Point sequences B15)
3.1.4. Real Variable Functions 216
1. Function of one real variable B16) 2 Functions of several real variables
B23).
3.1 5. Differentiation of functions of one real variable 225
1. Definition and geometric interpretation of the first derivative Examples B25) 2 Through
higher orders B26). 3. Properties of differentiable functions B27) 4 Monotonicity and convexity
functions B28). 5. Extrema and inflection points B29) 6 Elementary study of the ^ function
B30).
3.1.6. Differentiation of functions of several variables. N 2M
1. Partial derivatives, geometric interpretation B30) 2. Total differential, going through
direction, gradient B31) 3. Theorems on differentiable functions of several variables B32)
4. Differentiable mapping of the space Rn into Rm, functional definitions i el u. implicit
functions; existence theorems B33) 5 Change of variables in differential expressions
B35). 6. Extrema of functions of several variables B36)
3.1 7. Integral calculus of functions of one variable 238
1. Definite integrals B38) 2 Properties of definite integrals B39) 3 Indefinite
integrals B39). 4. Properties of indefinite integrals B41) 5 Integration of rational functions B42)
6. Integration of other classes of functions B44) 7 Improper intrals B47) 8 Geometic and
physical applications of definite integrals.B51)
3.1.8. Curvilinear integrals 253
1. Curvilinear integrals of the 1st kind (integrals over the length of the curve) B53) 2
calculation of curvilinear integrals of the 1st kind B53) 3 Curvilinear integrals of the 2nd kind
projection and general integrals) B54) 4. Properties and calculation of curvilinear integrals 2nd
genus B54). 5. Independence of the Curvilinear Integrals oi of the Integration Path B56) 6. Geometrical
and physical applications of curvilinear integrals B57)
3.1.9. Parameter Dependent Integrals 257
1. Definition of integral depending on parameter B57) 2 Properties of integrals depending on oi
parameter B57). 3. Improper integrals depending on parameter B58) 4 Examples of intrals,
depending on parameter B60)
3.1.10. Double integrals 2b0
1. Definition of the double integral and elementary properties B60) 2 Calculation of double integrals
B61). 3. Change of variables in double integrals B62) 4 Geometrical and physical applications
double integrals B63)
3.1.11. Triple integrals 263
1. Definition of the triple integral and elementary properties B63) 2 Calculation of triple hhicirals
B64). 3. Change of Variables in Triple Integrals B65). 4 Geometric and physical applications
triple integrals B65).
CONTENT
3.1.12. Surface integrals 266
1. Smooth surface area B66). 2. Surface integrals of the 1st and 2nd kind B66). 3. Geometric
and physical applications of the surface integral B69).
3.1.13. Integral Formulas 270
1. Ostrogradsky-Gauss formula. Green's formula B70). 2 Green's formulas B70). 3 Formula
Stokes B70). 4. Improper curvilinear - double, surface and triple integrals B70)
5. Multidimensional integrals depending on the parameter B72).
3.1.14. Endless rows 273
1. Basic concepts B73). 2. Tests for convergence or divergence of series with non-negative terms
B74). 3. Series with arbitrary members. Absolute convergence B76). 4 Functional
sequences. Function series B77). 5. Power Series B79). 6. Analytic functions. Taylor series.
Power series expansion of elementary functions B82).
3.1.15. Endless works 285
3.2. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
3.2.1. Calculus of Variations 287
1. Statement of the problem, examples and basic concepts B87). 2. Euler-Lagrange theory B88). 3.
Hamilton's theory - Jacobi B94). 4. Inverse problem of the calculus of variations B95). 5. Numerical methods
B95).
3.2.2. Optimal control 298
1. Basic concepts B98) 2. Pontryagin's maximum principle B98). 3. Discrete systems C03) 4.
Numerical methods C04).
3.3. DIFFERENTIAL URAV