Mathematical analysis. Mathematical analysis, functional analysis Mathematical analysis

M.: Moscow State University Publishing House. Part 1: 2nd ed., revised, 1985. - 662 pp.; Part 2- 1987. - 358 p.

Part 1. - Initial course.

The textbook represents the first part of a course in mathematical analysis for higher education. educational institutions USSR, Bulgaria and Hungary, written in accordance with the cooperation agreement between Moscow, Sofia and Budapest universities. The book includes the theory of real numbers, the theory of limits, the theory of continuity of functions, differential and integral calculus of functions of one variable and their applications, differential calculus of functions of several variables, and the theory of implicit functions.

Part 2. - Continuation of the course.

The textbook represents the second part (Part 1 - 1985) of a course in mathematical analysis, written in accordance with a unified program adopted in the USSR and the People's Republic of Belarus. The book covers the theory of numerical and functional series, the theory of multiple, curvilinear and surface integrals, field theory (including differential forms), the theory of parameter-dependent integrals, and the theory of Fourier series and integrals. The peculiarity of the book is three clearly separated levels of presentation: lightweight, basic and advanced, which allows it to be used both by students of technical universities with in-depth study of mathematical analysis, and by students of mechanical and mathematical faculties of universities.

Part 1. - Initial course.

Format: pdf

Size: 10.5 MB

Watch, download:drive.google

Format: djvu/zip

Size: 5.5 MB

/Download file

Part 2. - Continuation of the course.

Format: pdf

Size: 14.8 MB

Watch, download:drive.google

Format: djvu/zip

Size: 3.1 MB

/Download file

Part 1. - Initial course.

TABLE OF CONTENTS
Foreword by the title editor.... 5
Preface to the second edition 6
Preface to the first edition 6
Chapter 1. BASIC CONCEPTS OF MATHEMATICAL ANALYSIS 10
Chapter 2. REAL NUMBERS 29
§ 1. The set of numbers representable as infinite decimals, and its ordering 29
1. Properties of rational numbers (29). 2. Insufficiency of rational numbers for measuring segments of the number line (31). 3. Ordering a set of infinite decimals
fractions (34)
§ 2. Bounded above (or below) sets of numbers representable by infinite decimal fractions.... 40 1. Basic concepts (40). 2. Existence of exact edges (41).
§ 3. Approximation of numbers representable by infinite decimal fractions and rational numbers 44
§ 4. Operations of addition and multiplication. Description of the set of real numbers 46
1. Definition of addition and multiplication operations. Description of the concept of real numbers (46). 2. Existence and uniqueness of the sum and product of real numbers (47).
§ 5. Properties of real numbers 50
1. Properties of real numbers (50). 2. Some frequently used relationships (52). 3. Some concrete sets of real numbers (52).
§ 6. Additional questions in the theory of real numbers. .54 1. Completeness of the set of real numbers (54). 2. Axiomatic introduction of the set of real numbers (57).
§ 7. Elements of set theory. 59
1. The concept of set (59). 2. Operations on sets (60). 3. Countable and uncountable sets. Uncountable segment. Cardinality of the set (61). 4. Properties of operations on sets. Mapping sets (65).
Chapter 3. LIMIT THEORY. 68
§ 1. Sequence and its limit 68.
1. The concept of sequence. Arithmetic operations on sequences (68). 2. Bounded, unbounded, infinitesimal and infinitely large sequences (69). 3. Basic properties of infinitesimal sequences (73). 4. Converging sequences and their properties (75).
§ 2. Monotone sequences 83
1. The concept of a monotonic sequence (83). 2. Theorem on the convergence of a monotone bounded sequence (84). 3. Number e (86). 4. Examples of convergent monotone sequences (88).
§ 3. Arbitrary sequences 92
1. Limit points, upper and lower limits of the sequence (92). 2. Expansion of the concepts of limit point and upper and lower limits (99). 3. Cauchy criterion for the convergence of sequence (102).
§ 4. Limit (or limiting value) of a function 105
1. Concepts of variable quantity and function (105). 2. Limit of a function according to Heine and according to Cauchy (109). 3. Cauchy criterion for the existence of a limit of function (115). 4. Arithmetic operations on functions that have a limit (118). 5. Infinitely small and infinitely large functions (119).
§ 5. General definition limit of a function by base.... 122
Chapter 4. CONTINUITY OF FUNCTION 127
§ 1. The concept of continuity of a function 127
1. Definition of continuity of function (127). 2. Arithmetic operations on continuous functions (131). 3. Complex function and its continuity (132).
§ 2. Properties of monotone functions 132
1. Monotone functions (132). 2. The concept of an inverse function (133).
§ 3. The simplest elementary functions 138
1. Exponential function (138). 2. Logarithmic function (145). 3. Power function (146). 4. Trigonometric functions (147). 5. Inverse trigonometric functions (154). 6. Hyperbolic functions (156).
§ 4. Two remarkable limits 158
1. The first remarkable limit (158). 2. Second remarkable limit (159).
§ 5. Function discontinuity points and their classification. . . . 162 1. Classification of function discontinuity points (162). 2. On the discontinuity points of the monotone function (166).
§ 6. Local and global properties of continuous functions. 167 1. Local properties of continuous functions (167). 2. Global properties of continuous functions (170). 3. The concept of uniform continuity of a function (176). 4. The concept of modulus of continuity of a function (181).
§ 7. The concept of compactness of a set 184
1. Open and closed sets (184). 2. On coverings of a set by a system of open sets (184). 3. The concept of compactness of a set (186).
Chapter 5. DIFFERENTIAL CALCULUS 189
§ 1. The concept of derivative 189
1. Function increment. Difference form of the continuity condition (189). 2. Definition of derivative (190). 3. Geometric meaning of the derivative (192).
§ 2. The concept of differentiability of a function 193
1. Determination of differentiability of function (193). 2. Differentiability and continuity (195). 3. The concept of differential function (196).
§ 3. Differentiation complex function and inverse function 197 1. Differentiation of a complex function (197). 2. Differentiation of the inverse function (199). 3. Invariance of the form of the first differential (200). 4. Application of differential to establish approximate formulas (201).
§ 4. Differentiation of sum, difference, product and quotient of functions 202
§ 5. Derivatives of the simplest elementary functions. . . 205 1. Derivatives of trigonometric functions (205). 2. Derivative of the logarithmic function (207). 3. Derivatives of exponential and inverse trigonometric functions (208). 4. Derivative of power function (210). 5. Table of derivatives of the simplest elementary functions (210). 6. Table of differentials of the simplest elementary functions (212). 7. Logarithmic derivative. Derivative of power-exponential function (212).
§ 6. Derivatives and differentials of higher orders. . . 215 1. The concept of the lth order derivative (213). 2. nth derivatives of some functions (214). 3. Leibniz formula for the i-th derivative of the product of two functions (216). 4. Differentials of higher orders (218).
§ 7. Differentiation of a function given parametrically. 220*
§ 8. Derivative of a vector function 222
Chapter 6. BASIC THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS 224
§ 1. Increase (decrease) of a function at a point. Local extremum 224
§ 2. Theorem on the zero derivative 226
§ 3. Formula for finite increments (Lagrange formula). . 227 § 4. Some consequences from the Lagrange formula.... 229 "1. The constancy of a function that has a derivative (229) equal to zero on the interval. 2. Conditions for the monotonicity of a function on the interval (230). 3. Absence of discontinuities of the first kind and removable discontinuities in the derivative (231). 4. Derivation of some inequalities (233). § 5. Generalized formula for finite increments (Cauchy formula). . 234
§ 6. Disclosure of uncertainties (L'Hopital's rule). . . 235
1. Disclosure of uncertainty of the form (235). Disclosure of species uncertainty - (240). 3. Uncovering other types of uncertainties (243).
!§ 7. Taylor's formula “245
§ 8. Various shapes remainder member. Maclaurin formula 248
1. Residue term in Lagrange, Cauchy and Peano form (248).
2. Another entry for Taylor's formula (250). 3. Maclaurin formula (251).
§ 9. Estimation of the remainder term. Expansion of some elementary functions. . . . . 251
1. Estimation of the remainder term for an arbitrary function (251). 2. Expansion according to the Maclaurin formula of some elementary functions (252).
1§ 10. Examples of applications of the Maclaurin formula 256.
1. Calculation of the number e on a computer (256). 2. Proof of the irrationality of the number e (257). 3. Calculation of values of trigonometric functions (258). 4. Asymptotic estimation of elementary functions and calculation of limits (259).
Chapter 7. STUDYING THE GRAPH OF A FUNCTION AND FINDING EXTREME VALUES 262
§ 1. Finding stationary points 262
1. Signs of monotonicity of a function (262). 2. Finding stationary points (262). 3. The first sufficient condition for an extremum (264). 4. The second sufficient condition for an extremum "(265). 5. The third sufficient condition for an extremum (267). 6. The extremum of a function that is not differentiable at a given point (268). 7. The general scheme for finding extrema (270).
§ 2. Convexity of the graph of a function 271
§ 3. Inflection points 273
1. Determination of the inflection point. Prerequisite bend (273). 2. The first sufficient condition for inflection (276). 3. Some generalizations of the first sufficient condition for inflection (276). 4. Second sufficient condition for inflection (277). 5. Third sufficient condition for inflection (278).
§ 4. Asymptotes of the graph of a function 279
§ 5. Graphing a function 281
§ 6. Global maximum and minimum functions on a segment.
Edge extremum 284
1. Finding the maximum and minimum values of the function defined on the segment (284). 2. Edge extremum (286). 3. Darboux's theorem (287). Addition. An algorithm for finding extreme values of a function using only the values of that function. . . 288
Chapter 8. ANIMID FUNCTION AND INDEMNIFIC INTEGRAL 291
§ 1. The concept of a antiderivative function and the indefinite integral 291 1. The concept of a antiderivative function (291). 2. Indefinite integral (292). 3."Basic properties of the indefinite integral (293). 4. Table of basic indefinite integrals (294).
§ 2. Basic methods of integration 297
1, Integration of change of variable (substitution) (297).
2. Integration by parts (300).
§ 3. Classes of functions that are integrable in elementary functions. 303 1. Brief information about complex numbers (304). 2. Brief information about the roots of algebraic polynomials (307). 3. Decomposition of an algebraic polynomial with real coefficients into the product of irreducible factors (311). 4. Decomposition of a proper rational fraction into the sum of simple fractions (312). 5. Integrability of rational fractions in elementary functions (318). 6. Integrability in elementary functions of some trigonometric and irrational expressions (321).
§ 4. Elliptic integrals, 327
Chapter 9. RIEMANN DEFINITE INTEGRAL 330
§ 1. Definition of the integral. Integrability. . . . . 330 § 2. Upper and lower sums and their properties. . . . . 334 1. Determination of the upper and lower amounts (334). 2. Basic properties of upper and lower sums (335). § 3. Theorems on necessary and sufficient conditions for the integrability of functions. Classes of integrable functions. . . 339
1. Necessary and sufficient conditions for integrability (339).
2. Classes of integrable functions (341).
"§ 4. Properties of a definite integral. Estimates of integrals. Mean value theorems. 347
1. Properties of the integral (347). 2. Estimates of integrals (350).
§ 5. Antiderivative of a continuous function. Rules for integrating functions 357
1. Antiderivative (357). 2. Basic formula of integral calculus (359). 3. Important rules that allow you to calculate definite integrals (360). 4. The remainder of the Taylor formula in integral form (362).
§ 6. Inequalities for sums and integrals 365
1. Young's inequality (365). 2. Hölder's inequality for sums (366). 3. Minkowski inequality for sums (367). 4. Hölder’s inequality for integrals (367). 5. Minkowski inequality for integrals (368).
§ 7. Additional information about the definite Riemann integral 369
1. Limit of integral sums over the filter basis (369).
2. Lebesgue integrability criterion (370).
Appendix 1. Improper integrals 370
§ 1. Improper integrals of the first kind 371
1. The concept of an improper integral of the first kind (371).
2. Cauchy criterion for the convergence of an improper integral of the first kind. Sufficient signs of convergence (373). 3. Absolute and conditional convergence of improper integrals (375). 4. Change of variables under the improper integral sign and formula for integration by parts (378).
§ 2. Improper integrals of the second kind 379
§ 3. The main value of the improper integral.. 382
Appendix 2. Stieltjes integral 384
1. Definition of the Stieltjes integral and conditions for its existence (384). 2. Properties of the Stieltjes integral (389).
Chapter 10. GEOMETRICAL APPLICATIONS OF THE DETERMINATE INTEGRAL 391
§ 1. Arc length of a curve 391
1. The concept of a simple curve (391). 2. The concept of a parameterizable curve (392). 3. Length of the arc of the curve. The concept of a rectifiable curve (394). 4. Curve straightness criterion. Calculate the arc length of a curve (397). 5. Arc differential (402). 6. Examples (403).
!§ 2. Area of a flat figure 405
1. The concept of the boundary of a set and a plane figure (405).
2. Area of a flat figure (406). 3. Curvilinear area
trapezoid and curved sector (414). 4. Examples of calculating areas (416).
§ 3. Volume of a body in space 418
1. Body volume (418). 2. Some classes of cubed bodies (419). 3. Examples (421).
Chapter 11. APPROXIMATE METHODS FOR CALCULATING ROOTS OF EQUATIONS AND DETERMINED INTEGRALS... 422
§ 1. Approximate methods for calculating the roots of equations. . 422 1. Fork method (422). 2. Iteration method (423). 3. Methods of chords and tangents (426).
§ 2. Approximate methods for calculating definite integrals 431 1. Introductory remarks (431). 2. Rectangle method (434).
3. Trapezoidal method (436). 4. Parabola method (438).
Chapter 12. FUNCTIONS OF SEVERAL VARIABLES.... 442
§ 1. The concept of a function of m variables 442
1. The concept of m-dimensional coordinate and gameric Euclidean spaces (442). 2. Sets of points in m-dimensional Euclidean space (445). 3. The concept of a function of m variables (449).
§ 2. Limit of a function of m variables 451
1. Sequences of points in the space Em (451). 2. Property of a bounded sequence of points Em (454). 3. Limit of the function of m variables (455). 4. Infinitesimal functions of m variables (458). 5. Repeated limits (459).
§ 3. Continuity of a function of n variables 460
1. The concept of continuity of a function of m variables (460).
2. Continuity of a function of m variables in one variable (462). 3. Basic properties of continuous functions of several variables (465).
§ 4. Derivatives and differentials of functions of several variables 469
1. Partial derivatives of functions of several variables (469). 2. Differentiability of a function of several variables (470). 3. Geometric meaning of the condition for a differentiable function of two variables (473). 4. Sufficient conditions for differentiability (474). 5. Differential of a function of several variables (476). 6. Differentiation of complex functions (476). 7. Invariance of the form of the first differential (480). 8. Directional derivative. Gradient (481).
§ 5. Partial derivatives and differentials of higher orders.. 485 1. Partial derivatives of higher orders (485). 2. Differentials of higher orders (490). 3. Taylor's formula with a remainder term in Lagrange form and in integral form (497). 4. Taylor formula with a remainder term in Peano form (500).
6. Local extremum of a function of m variables.... 504 1. The concept of an extremum of a function of m variables. Necessary conditions for an extremum (504). 2. Sufficient conditions for a local extremum of a function of m variables (506). 3. The case of a function of two variables (512).
Appendix 1. Gradient method for searching for the extremum of a strongly convex function 514
1. Convex sets and convex functions (515). 2. The existence of a minimum for a strongly convex function and the uniqueness of a minimum for a strictly convex function (521).
3. Search for the minimum of a strongly convex function (526).
Appendix 2. Metric, normed spaces. . 535
Metric spaces. 1. Definition of metric space. Examples (535). 2. Open and closed sets (538). 3. Direct product of metric spaces (540). 4. Everywhere dense and perfect sets (541). 5. Convergence. Continuous displays (543). 6. Compactness (545). 7. Basis of space (548).
Properties of metric spaces 550
Topological spaces 558
1. Definition of topological space. Hausdorff topological space. Examples (558). 2. Remark on topological spaces (562).
Linear normed spaces, linear operators 564
1. Definition of linear space. Examples (564).
2. Normed spaces. Banach spaces.
Examples (566). 3. Operators in linear and normed spaces (568). 4. Space of operators (569).
5. Operator norm (569). 6. The concept of Hilbert space (572).
Appendix 3. Differential calculus in linear normed spaces. 574
1. The concept is differentiable. Strong and weak differentiability in linear normed spaces (575).
2. Lagrange formula for finite increments (581).
3. Relationship between weak and strong differentiability (584). 4. Differentiability of functionals (587). 5. Integral of abstract functions (587). 6. Newton-Leibniz formula for abstract functions (589). 7. Second order derivatives (592). 8. Mapping m-dimensional Euclidean space into g-dimensional space (595). 9. Derivatives and differentials of higher orders (598). 10. Taylor's formula for mapping one normed space into another (599).
Study on the extremum of functionals in normalized
spaces. 602
1. Necessary condition for extremum (602). 2. Sufficient conditions for an extremum (605).
Chapter 13. IMPLICIT FUNCTIONS 609
§ 1. Existence and differentiability of an implicitly given function 610
1. Theorem on the existence and differentiability of the implicit function (610). 2. Calculation of partial derivatives of an implicitly given function (615). 3. Singular points of a surface and a plane curve (617). 4. Conditions ensuring the existence of the inverse function (618) for the function y=)(x).
§ 2. Implicit functions defined by a system of functional
equations 619
1. Theorem on the solvability of a system of functional equations (619). 2. Calculation of partial derivatives of functions implicitly determined through a system of functional equations (624). 3. One-to-one mapping of two sets of m-dimensional space (625).
§ 3. Dependency of functions 626
1. The concept of function dependence. Sufficient condition for independence (626). 2. Functional matrices and their applications (628).
§ 4. Conditional extremum. 632
1. Concept conditional extremum(632). 2. Method of indefinite Lagrange multipliers (635). 3. Sufficient. conditions (636). 4. Example (637).
Appendix 1. Mappings of Banach spaces. Analog of the implicit function theorem 638
1. Theorem on the existence and differentiability of the implicit function (638). 2. The case of finite-dimensional spaces (644). 3. Singular points of a surface in a space of n dimensions. Reverse mapping (647). 4. Conditional extremum in the case of mappings of normed spaces (651).

Part 2. - Continuation of the course.

TABLE OF CONTENTS
Preface 5
CHAPTER 1. NUMBER SERIES 7
§ 1. The concept of a number series 7
1. Convergent and divergent series (7). 2. Cauchy criterion for the convergence of series (10)
§ 2. Series with non-negative terms 12"
1. Necessary and sufficient condition for the convergence of a series with non-negative terms (12). 2. Signs of comparison (13). 3. D'Alembert's and Cauchy's signs (16). 4. Integral Cauchy - MacLaurin test (21). 5, Raabe's sign (24). 6. Lack of a universal comparison series (27)
§ 3. Absolutely and conditionally convergent series 28
1. Concepts of absolutely and conditionally convergent series (28). 2. On the rearrangement of terms of the conditionally convergent series (30). 3. On the rearrangement of terms of an absolutely convergent series (33)
§ 4. Tests for the convergence of arbitrary series 35
§ 5. Arithmetic operations on convergent series 41
§ 6. Infinite products 44
1. Basic concepts (44). 2. Relationship between the convergence of infinite products and series (47). 3. Expansion of the function sin x into an infinite product (51)
§ 7. Generalized methods for summing divergent series.... 55
1. Cesaro's method (method of arithmetic averages) (56). 2. Poisson - Abel summation method (57)
§ 8. Elementary theory of double and repeated series 59
CHAPTER 2. FUNCTIONAL SEQUENCES AND SERIES 67
§ 1. Concepts of convergence at a point and uniform convergence on a set 67
1. The concepts of functional sequence and functional series (67). 2. Convergence of a functional sequence (functional series) at a point and on the set (69). 3. Uniform convergence on the set (70). 4. Cauchy criterion for uniform convergence of a sequence (series) (72)
§ 2. Sufficient criteria for uniform convergence of functional sequences and series 74
§ 3. Term-by-term passage to the limit 83
§ 4. Term-by-term integration and term-by-term differentiation of functional sequences and series 87
1. Term-by-term integration (87). 2. Term-by-term differentiation (90). 3. Convergence on average (94)
§ 5. Equivalent continuity of a sequence of functions... 97
§ 6. Power series 102
1. Power series and the region of its convergence (102). 2. Continuity of the sum of the power series (105). 3. Term-by-term integration and term-by-term differentiation of power series (105)
§ 7. Expansion of functions into power series 107
1. Expansion of a function into a power series (107). 2. Expansion of some elementary functions into Taylor series (108). 3. Elementary ideas about functions of a complex variable (CV). 4. Weierstrass’s theorem on uniform approximation of a continuous function by polynomials (112)
CHAPTER 3. DOUBLE AND n-MULTIPLE INTEGRALS 117
§ 1. Definition and conditions for the existence of a double integral. . . 117
1. Definition of double integral for a rectangle (117).
2. Conditions for the existence of a double integral for a rectangle (119). 3. Definition and conditions for the existence of a double integral for an arbitrary region (121). 4. General definition of double integral (123)
"§ 2. Basic properties of the double integral 127
§ 3. Reduction of a double integral to a repeated single integral. . . 129 1. The case of a rectangle (129). 2. The case of an arbitrary region (130)
§ 4. Triple and n-fold integrals 133
§ 5. Change of variables in an n-fold integral 138
§ 6. Calculation of volumes of n-dimensional bodies 152
§ 7. Theorem on term-by-term integration of functional sequences and series 157
$ 8. Multiple improper integrals 159
1. The concept of multiple improper integrals (159). 2. Two criteria for the convergence of improper integrals of nonnegative functions (160). 3. Improper integrals of alternating functions (161). 4. Principal value of multiple improper integrals (165)
CHAPTER 4. CURVILINEAR INTEGRALS 167
§ 1. Concepts of curvilinear integrals of the first and second kind. . . 167
§ 2. Conditions for the existence of curvilinear integrals 169
CHAPTER 5. SURFACE INTEGRALS 175
§ 1. Concepts of surface and its area 175
1. The concept of surface (175). 2. Auxiliary lemmas (179).
3. Surface area (181)
§ 2. Surface integrals 185
CHAPTER 6. FIELD THEORY. BASIC INTEGRAL FORMULAS OF ANALYSIS 190
§ 1. Notation. Biorthogonal bases. Invariants of the linear operator 190
1. Notation (190). 2. Biorthogonal bases in the space E" (191). 3. Transformations of bases. Covariant and contravariant coordinates of a vector (192). 4. Invariants of a linear operator. Divergence and curl (195). 5. Expressions for divergence and curl of a linear operator in an orthonormal basis (Shch8)
§ 2. Scalar and vector fields. Differential operators of vector analysis 198
!. Scalar and vector fields (198). 2. Divergence, rotor and derivative with respect to the direction of a vector field (203). 3. Some other vector analysis formulas (204). 4. Concluding remarks (206)
§ 3. Basic integral formulas of analysis 207
1. Green's formula (207). 2. Ostrogradsky-Gauss formula (211). 3. Stokes formula (214)
§ 4. Conditions for the independence of a curvilinear integral on the plane of the integration path 218
§ 5. Some examples of applications of field theory 222
1. Expression of the area of a flat region through a curvilinear integral (222). 2. Expression of volume through surface integral (223)
Supplement to Chapter 6. Differential forms in Euclidean space 225
§ 1. Alternating multilinear forms 225
1. Linear forms (225). 2. Bilinear forms (226). 3. Multilinear forms (227). 4. Alternating polylinear forms (228). 5. External product of alternating forms (228). 6. Properties of the outer product of alternating forms (231). 7. Basis in the space of alternating forms (233)
§ 2. Differential forms 235
1. Basic notations (235). 2. External differential (236). 3. Properties of external differential (237;)
§ 3. Differentiable mappings 2391
1. Definition of differentiable mappings (239). 2. Display properties f* (240)
§ 4. Integration of differential forms 243
1. Definitions (243). 2. Differentiable chains (245). 3. Stokes formula (248). 4. Examples (250)
CHAPTER 7. INTEGRALS DEPENDING ON PARAMETERS 252
§ 1. Uniform in one variable the tendency of a function of two variables to the limit in another variable 252
1. The connection between a function of two variables tending uniformly in one variable to the limit in another variable with the uniform convergence of the functional sequence (252). 2. Cauchy criterion for the uniform tendency of a function to the limit (254). 3. Applications of the concept of uniform tendency to the limit function (254)
§ 2. Proper integrals depending on the parameter 256
1. Properties of the integral depending on the parameter (256). 2. The case when the limits of integration depend on the parameter (257)
§ 3. Improper integrals depending on the parameter 259
1. Improper integrals of the first kind, depending on the parameter (260). 2. Improper integrals of the second kind depending on the parameter (266)
§ 4. Application of the theory of integrals depending on a parameter to the calculation of some improper integrals 267
§ 5. Euler integrals 271
k G-function (272). 2. B-function (275). 3. Relationship between Euler integrals (277). 4. Examples (279)
§ 6. Stirling formula 280
§ 7. Multiple integrals depending on parameters 282
1. Own multiple integrals depending on parameters (282).
2. Improper multiple integrals depending on parameter (283)
CHAPTER 8. FOURIER SERIES 287
§ 1. Orthonormal systems and general Fourier series 287
1. Orthonormal systems (287). 2. The concept of a general Fourier series (292)
§ 2. Closed and complete orthonormal systems 295
§ 3. Closedness of the trigonometric system and consequences from it. . 298 1. Uniform approximation of a continuous function by trigonometric polynomials (298). 2. Proof of the closedness of the trigonometric system (301). 3. Consequences of the closedness of the trigonometric system (303)
§ 4. The simplest conditions for uniform convergence and term-by-term differentiation of a trigonometric Fourier series 304
1. Introductory remarks (304). 2. The simplest conditions for absolute and uniform convergence of the trigonometric Fourier series (306).
3. The simplest conditions for term-by-term differentiation of the trigonometric Fourier series (308)
§ 5. More precise conditions for uniform convergence and conditions for convergence at a given point 309>
1. Modulus of continuity of a function. Hölder classes (309). 2. Expression for the partial sum of the trigonometric Fourier series (311). 3. Auxiliary sentences (314). 4. The principle of localization (317). 5. Uniform convergence of the trigonometric Fourier series for a function from the Hölder class (319). 6. On the convergence of the trigonometric Fourier series of the piecewise Hölder function (325). 7. Summability of the trigonometric Fourier series of a continuous function by the method of arithmetic means (329). 8. Concluding remarks (331)
§ 6. Multiple trigonometric Fourier series 332
1. Concepts of multiple trigonometric Fourier series and its rectangular and spherical partial sums (332). 2. Modulus of continuity and Hölder classes for a function of N variables (334). 3. Conditions for the absolute convergence of a multiple trigonometric Fourier series (335)
CHAPTER 9. FOURIER TRANSFORM 33"
§ 1. Representation of a function by the Fourier integral 339
1. Auxiliary statements (340). 2. Main theorem. Inversion formula (342). 3. Examples (347)
§ 2. Some properties of the Fourier transform 34&
§ 3. Multiple Fourier integral 352

Aleksic G. Problems of convergence of orthogonal series. M.: IL, 1963 (djvu)
Akhiezer N., Krein M. On some issues in the theory of moments. Kharkov: GNTIU, 1938 (djvu)
Akhiezer N.I. The classical problem of moments and some questions of analysis related to it. M.: Fizmatlit, 1961 (djvu)
Balk M.B., Petrov V.A., Polukhin A.A. A workbook on the theory of analytic functions. M.: Education, 1976 (djvu)
Beckenbach E., Bellman R. Introduction to inequalities. M,: Mir, 1965 (djvu)
Bernstein S.N. Extremal properties of polynomials and best approximation of continuous functions of one real variable. Part 1. Leningrad-M.: GROTL, 1937 (djvu)
Bermant A.F. Course of mathematical analysis. Part I (12th ed.). M. Fizmatgiz, 1959 (djvu)
Bermant A.F. Course of mathematical analysis. Part II (9th ed.). M. Fizmatgiz, 1959 (djvu)
Bermant A.F., Aramanovich I.G. A short course in mathematical analysis for technical colleges (5th ed.). M.: Nauka, 1967 (djvu)
Brelo M. On topologies and boundaries in potential theory. M.: Mir, 1974 (djvu)
Brudno A.L. Theory of functions of a real variable. M.: Nauka, 1971 (djvu)
Budak B.M., Fomin S.V. Multiple integrals and series. M.: Nauka, 1965 (djvu)
Budylin A.M. Fourier series and integrals. L.: St. Petersburg State University, 2002 (pdf)
Bourbaki N. Functions of a real variable. Elementary theory. M.: Nauka, 1965 (djvu)
Baer R. Theory of discontinuous functions. M.-L.: GTTIL, 1932 (djvu)
Vallee-Poussin C.-J. Course of analysis of infinitesimals, volume 1. 1922 (djvu)
Vallee-Poussin C.-J. Course of analysis of infinitesimals, volume 2. L.-M.: GTTI, 1933 (djvu)
Vilenkin N.Ya., Bokhan K.A., Maron I.A., Matveev I.V., Smolyansky M.L., Tsvetkov A.T. Problem book for the course of mathematical analysis. Part I. M.: Enlightenment, 1971 (djvu)
Vilenkin N.Ya., Bokhan K.A., Maron I.A., Matveev I.V., Smolyansky M.L., Tsvetkov A.T. Problem book for the course of mathematical analysis. Part II. M.: Education, 1971 (djvu)
Vulikh B.Z. Introduction to Functional Analysis (2nd ed.). M.: Nauka, 1967 (djvu)
Vulikh B.Z. A short course on the theory of functions of a real variable. Introduction to Integral Theory (2nd ed.). M.: Nauka, 1973 (djvu)
Vygodsky M.Ya. Handbook of Higher Mathematics (12th ed.). M.: Nauka, 1977 (djvu)
Vygodsky M.Ya. Fundamentals of Infinitesimal Calculus (3rd ed.). M.-L.: GTTI, 1933 (djvu)
Hardy G. Integration of elementary functions. M.-L.: ONTI, 1935 (djvu)
Gelbaum B., Olmstead J. Counterexamples in analysis. M.: Mir, 1967 (djvu)
Gelfand I.M., Vilenkin N.Ya. Some applications of harmonic analysis. Framed Hilbert spaces. (Generalized Functions, Issue 4). M.: Fizmatlit, 1961 (djvu)
Gelfand I.M., Graev M., Vilenkin N.Ya. Integral geometry and related issues of representation theory. (Generalized Functions, Issue 5). M.: Fizmatlit, 1962 (djvu)
Gelfand I.M., Graev M., Pyatetsky-Shapiro I. Representation theory and automorphic functions (Generalized functions, issue 6). M.: Fizmatlit, 1966 (djvu)
Gelfand I.M., Raikov D.A., Shilov G.E. Commutative normed rings. M.: GIFML, 1960 (djvu)
Gelfand I.M., Shilov G.E. Generic Functions and Operations on Them (Generic Functions, Issue 1) (2nd ed.). M.: Fizmatlit, 1959 (djvu)
Gelfand I.M., Shilov G.E. Spaces of basic and generalized functions (Generalized Functions, Issue 2). M.: Fizmatlit, 1958 (djvu)
Gelfand I.M., Shilov G.E. Some questions in the theory of differential equations (Generalized functions, issue 3). M.: Fizmatlit, 1958 (djvu)
Glivenko V.I. Stieltjes integral. L.: ONTI, 1936 (djvu)
Gradshtein I.S. Ryzhik I.M. Tables of integrals, sums, series and products (4th ed.). M.: Nauka, 1963 (djvu)
Gursa E. Course of mathematical analysis, volume 1, part 1. Derivatives and differentials. Definite integrals. M.-L.: GTTI, 1933 (djvu)
Gursa E. Course of mathematical analysis, volume 1, part 2. Series expansions. Geometric applications. M.-L.: GTTI, 1933 (djvu)
Gursa E. Course of mathematical analysis, volume 2, part 1. Theory of analytic functions. M.-L.: GTTI, 1933 (djvu)
Gursa E. Course of mathematical analysis, volume 2, part 2. Differential equations. M.-L.: GTTI, 1933 (djvu)
Gursa E. Course of mathematical analysis, volume 3, part 1. Infinitely close integrals. Partial differential equations. M.-L.: GTTI, 1933 (djvu)
Gursa E. Course of mathematical analysis, volume 3, part 2. Integral equations. Calculus of variations. M.-L.: GTTI, 1934 (djvu)
De Bruijn N.G. Asymptotic methods in analysis. M.: IL, 1961 (djvu)
De Rham J. Differentiable varieties. M.: IL, 1956 (djvu)
Davydov N.A., Korovkin P.P., Nikolsky V.N. Collection of problems in mathematical analysis (4th ed.). M.: Education, 1973 (djvu)
Demidovich B.P. (ed.). Problems and exercises in mathematical analysis for technical colleges (6th ed.). M.: Nauka, 1968 (djvu)
Demidovich B.P. (ed.) Problems and exercises in mathematical analysis for technical colleges (10th ed.). M.: Nauka, 1978 (djvu)
Demidovich B.P. Collection of problems and exercises in mathematical analysis. M.: Nauka, 1966 (djvu)
Demidov A.S. Generalized Functions in Mathematical Physics: Main Ideas and Concepts. New York: Nova Science, 2001 (pdf)
Jackson D. Fourier series and orthogonal polynomials. M.: IL, 1948 (djvu)
Jenkins G., Watts D. Spectral analysis and its applications. Issue 1. M.: Mir, 1971 (djvu)
Jenkins G., Watts D. Spectral analysis and its applications. Issue 2. M.: Mir, 1972 (djvu)
Dieudonne J. Fundamentals of modern analysis. M.: Mir, 1964 (djvu)
Egorova I.A. A workbook on mathematical analysis. Part III. Functions of several variables. M.: Uchpedgiz, 1962 (djvu)
Erugin N.P. Implicit functions. L.: Leningrad State University, 1956 (djvu)
Zaporozhets G.I. A guide to solving problems in mathematical analysis (4th ed.). M.: Higher School, 1966 (djvu)
Zeldovich B., Myshkis A.D. Elements of Applied Mathematics (3rd ed.). M.: Nauka, 1972 (djvu)
Zeldovich Ya.B., Yaglom I.M. Higher mathematics for beginning physicists and technicians. M.: Nauka, 1982 (djvu)
Zygmund A. Trigonometric series, volume 1. M.: Mir, 1965 (djvu)
Zygmund A. Trigonometric series, volume 2. M.: Mir, 1965 (djvu)
Yoshida K. Functional analysis. M.: Mir, 1967 (djvu)
Kazimirov N.I. Mathematical analysis. Lecture notes for the first year, PetrSU (pdf)
Kalinin V.V., Petrova I.V., Kharin V.T. Indefinite and definite integrals (Mathematics in oil and gas education, issue 3, part 1). M.: MGUNG im. THEM. Gubkina, 2005 (pdf)
Kamke E. Lebesgue-Stieltjes integral. M.: Fizmatlit, 1959 (djvu)
Kaplan I.A. Practical classes in higher mathematics. Parts 1, 2, 3. Analytical geometry on the plane and in space. Differential calculus of functions of one and many independent variables. Integral calculus of functions of one independent variable, integration of differential equations (3rd ed.). Kharkov: KhSU, 1967 (djvu)
Kaplan I.A. Practical classes in higher mathematics. Part II. Differential calculus of functions of one and many independent variables (5th ed.). Kharkov: Vishcha School, 1973 (djvu)
Kaplan I.A. Practical classes in higher mathematics. Part III. Integral calculus of a function of one independent variable. Integration of Differential Equations (4th ed.). Kharkov: Vishcha School, 1974 (djvu)
Kaplan I.A. Practical classes in higher mathematics. Part IV. Multiple and curvilinear integrals (2nd ed.). Kharkov: KhSU, 1971 (djvu)
Kaplan I.A. Practical classes in higher mathematics. Part V. Numerical solution of algebraic and transcendental equations, matrix calculus, vector analysis and integration of first order linear partial differential equations. (2nd ed.). Kharkov: KhSU, 1972 (djvu)
Karlin S., Stadden V. Chebyshev systems and their application in analysis and statistics. M.: Nauka, 1976 (djvu)
Cartan A. Differential calculus. Differential forms. M.: Mir, 1971 (djvu) (djvu)
Kachenovsky M.I., Bokhan K.M., Karpenko K.M. Collection tests in mathematical disciplines. Issue I. M.: Uchpedgiz, 1958 (djvu)
Kozhevnikov N.I., Krasnoshchekova T.I., Shishkin N.E. Series and Fourier integral. Field theory. Analytical and special functions. Laplace transform. M.: Nauka, 1964 (djvu)
Collatz L. Functional analysis and computational mathematics. M.: Mir, 1969 (djvu)
Kolmogorov A.N., Fomin S.V. Elements of function theory and functional analysis (4th ed.). M.: Nauka, 1976 (djvu)
Copson E.T. Asymptotic expansions. M.: Mir, 1966 (djvu)
Korn G., Korn T. Handbook of mathematics for scientists and engineers. M.: Nauka, 1973 (djvu)
Korobov N.M. Number-theoretic methods in approximate analysis. M.: Fizmatlit, 1963 (djvu)
Cauchy G.A.L. Differential and integral calculus. St. Petersburg: Imperial Academy of Sciences, 1831 (djvu)
Krein S.G., Ushakova V.N. Mathematical analysis of elementary functions. M.: GIFML, 1963 (djvu)
Kurant R. Course of differential and integral calculus, volume 1. M.: Nauka, 1967 (djvu)
Kurant R. Course of differential and integral calculus, volume 2. M.: Nauka, 1970 (djvu)
Kushner B.A. Lectures on constructive mathematical analysis. M.: Nauka, 1973 (djvu)
Landau E. Fundamentals of analysis. M.: IL, 1947 (djvu)
Laschenov K.V. A workbook on mathematical analysis. Integral calculus of functions of one variable. M.: Uchpedgiz, 1963 (djvu)
Lebesgue A. Integration and finding primitive functions. M.-L.: GTTI, 1934 (djvu)
Levitan B.M. Almost periodic functions. M.: GITTL, 1953 (djvu)
Levitan B.M., Zhikov V.V. Almost periodic functions and differential equations. M.: MSU, 1978 (djvu)
Lang S. Introduction to the theory of differentiable manifolds. M.: Mir, 1967 (djvu)
Lefort G. Algebra and analysis. Tasks. M.: Nauka, 1973 (djvu)
Likholetov I.I., Matskevich I.P. A guide to solving problems in higher mathematics, probability theory and mathematical statistics (2nd ed.). Mn.: Vysh. school, 1969 (djvu)
L'Hopital G.F. Analysis of infinitesimals. M.-L.: Gostekhteorizdat, 1935 (djvu)
Luzin N.N. Differential calculus (7th ed.). M.: Higher. school, 1961 (djvu)
Luzin N.N. Integral and trigonometric series. M.-L.: GITTL, 1951 (djvu)
Luzin N.N. Integral calculus (7th ed.). M.: Higher. school, 1961 (djvu)
Luzin N.N. On some new results of the descriptive theory of functions. M.-L.: USSR Academy of Sciences, 1935 (djvu)
Luzin N.N. Current state theory of functions of a real variable. M.-L.: GTTI, 1933 (djvu)
Lumis L. Introduction to abstract harmonic analysis. M.: IL, 1956 (djvu)
Lyusternik L.A., Sobolev V.I. Elements of the functional (2nd ed.). M.: Nauka, 1965 (djvu)
McDonald I. Symmetric functions and Hall polynomials. M.: Mir, 1972 (djvu)
Malgrange B. Ideals of differentiable functions. M.: Mir, 1968 (djvu)
Maron I.A. Differential and integral calculus in examples and problems. Functions of one variable. M.: Nauka, 1970 (djvu)
Myshkis A.D. Lectures on higher mathematics (4th ed.). M.: Nauka, 1973 (djvu)
Myshkis A.D. Mathematics for colleges. Special courses. M.: Nauka, 1971 (djvu)
Narasimhan R. Analysis on real and complex manifolds. M.: Mir, 1971 (djvu)
Nathanson I.P. Constructive theory of functions. M.-L.: GITTL, 1949 (djvu)
Nathanson I.P. Theory of functions of a real variable. M.-L.: GITTL, 1950 (djvu)
Nathanson I.P. Theory of functions of a real variable (3rd ed.). M.: Nauka, 1974 (djvu)
Nezbaylo T.G. New theory for calculating the indefinite integral. St. Petersburg: Corona-Vek, 2007 (pdf)
Nemytsky V., Sludskaya M., Cherkasov A. Course in mathematical analysis. Volume I. M.-L.: GITTL, 1940 (djvu)
Ochan Yu.S. Collection of problems and theorems on the theory of functions of a real variable. M.: Education, 1963 (djvu)
Parfentiev N.N. Research on the theory of growth of functions. Kazan, KazUn, 1910 (djvu)
Pogorelov A.I. Tests on mathematical analysis. M.: Uchpedgiz, 1951 (djvu)
Pogorelov A.I. Collection of problems in higher mathematics. M.: Uchpedgiz, 1949 (djvu)
Polya G., Szege G. Problems and theorems from analysis. Part 1. Rows. Integral calculus. Theory of functions. M.: Nauka, 1978 (djvu)
Polya G., Szege G. Problems and theorems from analysis. Part 2. Theory of functions. Distribution of zeros. Polynomials. Determinants. Number theory. M.: Nauka, 1978 (djvu)
Riekstynysh E.Ya. Asymptotic expansions of integrals. Volume 1. Riga: Zinatne, 1974 (djvu)
Riekstynysh E.Ya. Asymptotic expansions of integrals. Volume 2. Riga: Zinatne, 1977 (djvu)
Riekstynysh E.Ya. Asymptotic expansions of integrals. Volume 3. Riga: Zinatne, 1981 (djvu)
Rudin U. Fundamentals of mathematical analysis (2nd ed.). M.: Mir, 1976 (djvu)
Ryvkin A.Z., Kunitskaya E.S. A workbook on mathematical analysis. Part 2. Integral calculus of functions of one variable. M.: Uchpedgiz, 1962 (djvu)
Sachs S. Integral theory. M.: IL, 1949 (djvu)
Collection of tests in mathematical disciplines (for part-time students who graduated from teacher training institutes). M.: Uchpedgiz, 1958 (djvu)
Sviridyuk G.A., Fedorov V.E. Mathematical analysis. Part 1. Chelyabinsk: ChelSU, 1999 (pdf)
Sviridyuk G.A., Kuznetsov G.A. Mathematical analysis. Part 2. Chelyabinsk: ChelSU, 1999 (pdf)
Smirnov V.I. Course of higher mathematics, volume 1 (23rd edition). M.: Nauka, 1974 (djvu)
Smirnov V.I. Course of higher mathematics, volume 2 (21st edition). M.: Nauka, 1974 (djvu)
Smirnov V.I. Course of higher mathematics, volume 3, part 1 (10th edition). M.: Nauka, 1974 (djvu)
Smirnov V.I. Course of higher mathematics, volume 3, part 2 (9th edition). M.: Nauka, 1974 (djvu)
Smirnov V.I. Course of higher mathematics, volume 4, part 1 (6th edition). M.: Nauka, 1974 (djvu)
Smirnov V.I. Course of higher mathematics, volume 4, part 2 (6th edition). M.: Nauka, 1981 (djvu)
Smirnov V.I. Course of higher mathematics, volume 5. M.: GIFML, 1959

Zorich V. A. Mathematical analysis. Part I. – Ed. 4th, rev. – M.: MTsNMO, 2002. – XVI + 664 p.

Zorich V. A. Mathematical analysis. Part II. – Ed. 4th, rev. – M.: MTsNMO, 2002. – XIV + 794 p.

University textbook in two volumes for students of physics and mathematics. It may be useful for students of faculties and universities with advanced mathematical training, as well as specialists in the field of mathematics and its applications.

The book reflects the connection between the course of classical analysis and modern mathematical courses (algebra, differential geometry, differential equations, complex and functional analysis).

Main sections of the first part: introduction to analysis (logical symbolism, set, function, real number, limit, continuity); differential and integral calculus of a function of one variable; differential calculus of functions of several variables.

The second part of the textbook includes the following sections: Multidimensional integral. Differential forms and their integration. Series and integrals depending on a parameter (including series and Fourier transforms, as well as asymptotic expansions).

Part I

Chapter I. Some general mathematical concepts and notations
- § 1. Logical symbolism
  - 1. Ligaments and brackets.
  - 2. Notes on evidence.
  - 3. Some special notations.
  - 4. Concluding remarks.
- § 2. Set and elementary operations on sets
  - 1. The concept of set.
  - 2. Inclusion relation.
  - 3. The simplest operations on sets.
- § 3. Function
  - 1. The concept of a function (mapping).
  - 2. The simplest classification of mappings.
  - 3. Composition of functions and mutually inverse mappings.
  - 4. Function as a relation. Function graph.
- § 4. Some additions
  - 1. Cardinality of the set (cardinal numbers).
  - 2. On the axiomatics of set theory.
  - 3. Remarks on the structure of mathematical statements and their writing in the language of set theory.
Chapter II. Real numbers
- § 1. Axiomatics and some general properties of the set of real numbers
  - 1. Definition of the set of real numbers.
  - 2. Some general algebraic properties of real numbers.
  - 3. Axiom of completeness and the existence of an upper (lower) bound of a numerical set.
- § 2. The most important classes of real numbers and computational aspects of operations with real numbers
  - 1. Natural numbers and the principle of mathematical induction.
  - 2. Rational and irrational numbers.
  - 3. Archimedes' principle.
  - 4. Geometric interpretation of the set of real numbers and computational aspects of operations with real numbers.
- § 3. Basic lemmas related to the completeness of the set of real numbers
  - 1. Lemma on nested segments (Cauchy-Cantor principle).
  - 2. Finite covering lemma (Borel-Lebesgue principle.
  - 3. Lemma on the limit point (Bolzano-Weierstrass principle).
- § 4. Countable and uncountable sets
  - 1. Countable sets.
  - 2. Continuum power.
Chapter III. Limit
- § 1. Sequence limit
  - 1. Definitions and examples.
  - 2. Properties of the sequence limit.
  - 3. Questions about the existence of a sequence limit.
  - 4. Initial information about the series.
- § 2. Limit of a function
  - 1. Definitions and examples.
  - 2. Properties of the limit of a function.
  - 3. General definition of the limit of a function (limit by base).
  - 4. Questions about the existence of the limit of a function.
Chapter IV. Continuous functions
- § 1. Basic definitions and examples
  - 1. Continuity of a function at a point.
  - 2. Breaking points.
- § 2. Properties of continuous functions
  - 1. Local properties.
  - 2. Global properties of continuous functions.
Chapter V. Differential calculus
- § 1. Differentiable function
  - 2. Function differentiable at a point.
  - 3. Tangent; geometric meaning derivative and differential.
  - 4. The role of the coordinate system.
  - 5. Some examples.
- § 2. Basic rules of differentiation
  - 1. Differentiation and arithmetic operations.
  - 2. Differentiation of the composition of functions.
  - 3. Differentiation of the inverse function.
  - 4. Table of derivatives of basic elementary functions.
  - 5. Differentiation of the simplest implicitly given function.
  - 6. Higher order derivatives.
- § 3. Basic theorems of differential calculus
  - 1. Fermat's lemma and Rolle's theorem.
  - 2. Lagrange and Cauchy theorems on finite increment.
  - 3. Taylor's formula.
- § 4. Study of functions by methods of differential calculus
  - 1. Conditions for a function to be monotonic.
  - 2. Conditions for the internal extremum of the function.
  - 3. Conditions for the convexity of a function.
  - 4. L'Hopital's rule.
  - 5. Graphing a function.
- § 5. Complex numbers and the relationship of elementary functions 2
  - 1. Complex numbers.
  - 2. Convergence in C and series with complex terms.
  - 3. Euler's formula and the relationship of elementary functions.
  - 4. Representation of a function by a power series, analyticity.
  - 5. Algebraic closedness of the field C of complex numbers.
- § 6. Some examples of the use of differential calculus in problems of natural science
  - 1. Movement of a body of variable mass.
  - 2. Barometric formula.
  - 3. Radioactive decay, chain reaction and a nuclear boiler.
  - 4. Falling bodies in the atmosphere.
  - 5. Once again about the number e and functions.
  - 6. Oscillations.
- § 7. Antiderivative
  - 1. Antiderivative and indefinite integral.
  - 2. Basic general techniques for finding an antiderivative.
  - 3. Antiderivatives of rational functions.
  - 4. Prototypes.
  - 5. Prototypes.
Chapter VI. Integral
- § 1. Definition of the integral and description of the set of integrable functions
  - 1. Objective and guiding considerations.
  - 2. Definition of the Riemann integral.
  - 3. Many integrated functions.
- § 2. Linearity, additivity and monotonicity of the integral
  - 1. Integral as a linear function on space.
  - 2. Integral as an additive function of the segment of integration.
  - 3. Estimation of the integral, monotonicity of the integral, mean value theorems.
- § 3. Integral and derivative
  - 1. Integral and antiderivative.
  - 2. Newton-Leibniz formula.
  - 3. Integration by parts in a definite integral and Taylor's formula.
  - 4. Change of variable in the integral.
  - 5. Some examples.
- § 4. Some applications of the integral
  - 1. Additive function of oriented interval and integral.
  - 2. Path length.
  - 3. Area of a curved trapezoid.
  - 4. Volume of a body of revolution.
  - 5. Work and energy.
- § 5. Improper integral
  - 1. Definitions, examples and basic properties of improper integrals.
  - 2. Study of the convergence of the improper integral.
  - 3. Improper integrals with several singularities.
Chapter VII. Functions of several variables, their limit and continuity
- § 1. The space R m and the most important classes of its subsets
  - 1. The set R m and the distance in it.
  - 2. Open and closed sets in R m.
  - 3. Compacta in R m.
  - Tasks and exercises.
- § 2. Limit and continuity of a function of several variables
  - 1. Function limit.
  - 2. Continuity of functions of several variables and properties of continuous functions.
Chapter VIII. Differential calculus of functions of several variables
- § 1. Linear structure in R m
  - 1. R m as a vector space.
  - 2. Linear mappings.
  - 3. Norm in R m.
  - 4. Euclidean structure in R m.
- § 2. Differential of a function of several variables
  - 1. Differentiability and differential of a function at a point.
  - 2. Differential and partial derivatives of a real-valued function.
  - 3. Coordinate representation of the mapping differential. Jacobian matrix.
  - 4. Continuity, partial derivatives and differentiability of a function at a point.
- § 3. Basic laws of differentiation
  - 1. Linearity of the differentiation operation.
  - 2. Differentiation of the composition of mappings.
  - 3. Differentiation of the inverse mapping.
- § 4. Basic facts of differential calculus of real-valued functions of several variables
  - 1. Mean value theorem.
  - 2. Sufficient condition for differentiability of a function of several variables.
  - 3. Higher order partial derivatives.
  - 4. Taylor's formula.
  - 5. Extrema of functions of several variables.
  - 6. Some geometric images associated with functions of many variables.
- § 5. Implicit function theorem
  - 1. Statement of the question and guiding considerations.
  - 2. The simplest option implicit function theorems.
  - 3. Transition to the case of dependence F(x 1, ..., x n, y) = 0.
  - 4. Implicit function theorem.
- § 6. Some consequences of the implicit function theorem
  - 1. Theorem about the inverse function.
  - 2. Local reduction of a smooth map to canonical form.
  - 3. Dependency of functions.
  - 4. Local decomposition of a diffeomorphism into a composition of protozoa.
  - 5. Morse Lemma.
- § 7. Surface in R n and the theory of conditional extremum
  - 1. Surface of dimension k in Rn.
  - 2. Tangent space.
  - 3. Conditional extremum.
Some tasks of colloquiums
Questions for the exam
Literature
Alphabetical index

Part II

Chapter IX. Continuous mappings (general theory)
- § 1. Metric space
  - 1. Definitions and examples.
  - 2. Open and closed subsets of a metric space.
  - 3. Subspace of metric space.
  - 4. Direct product of metric spaces.
- § 2. Topological space
  - 1. Basic definitions.
  - 2. Subspace of topological space.
  - 3. Direct product of topological spaces.
- § 3. Compacts
  - 1. Definition and general properties of a compact.
  - 2. Metric compacts.
- § 4. Connected topological spaces
- § 5. Complete metric spaces
  - 1. Basic definitions and examples.
  - 2. Replenishment of metric space.
- § 6. Continuous maps of topological spaces
  - 1. Display limit.
  - 2. Continuous mappings.
- § 7. The principle of contraction mappings
Chapter X. Differential calculus from a more general point of view
- § 1. Linear normed space
  - 1. Some examples of linear spaces of analysis.
  - 2. Norm in vector space.
  - 3. Scalar product in vector space.
- § 2. Linear and multilinear operators
  - 1. Definitions and examples.
  - 2. Operator norm.
  - 3. Space of continuous operators.
- § 3. Mapping differential
  - 1. Map differentiable at a point.
  - 2. General laws of differentiation.
  - 3. Some examples.
  - 4. Partial derivatives of mappings.
- § 4. The finite increment theorem and some examples of its use
  - 1. Finite increment theorem.
  - 2. Some examples of the application of the finite increment theorem.
- § 5. Derivative mappings of higher orders
  - 1. Definition of the nth differential.
  - 2. Vector derivative and calculation of the values of the nth differential.
  - 3. Symmetry of higher order differentials.
  - 4. Some comments.
- § 6. Taylor's formula and the study of extrema
  - 1. Taylor's formula for mappings.
  - 2. Study of internal extrema.
  - 3. Some examples.
- § 7. General theorem on the implicit function
Chapter XI. Multiple integrals
- § 1. Riemann integral on an n-dimensional interval
  - 1. Definition of the integral.
  - 2. Lebesgue criterion for the integrability of a function according to Pnmann.
  - 3. Darboux criterion.
- § 2. Integral over a set
  - 1. Admissible sets.
  - 2. Integral over a set.
  - 3. Measure (volume) of an admissible set.
- § 3. General properties of the integral
  - 1. Integral as a linear functional.
  - 2. Additivity of the integral.
  - 3. Estimates of the integral.
- § 4. Reduction of a multiple integral to a repeated one
  - 1. Fubini's theorem.
  - 2. Some consequences.
- § 5. Change of variables in a multiple integral 139
  - 1. Statement of the question and heuristic derivation of the formula - replacement of variables.
  - 2. Measurable sets and smooth mappings.
  - 3. One-dimensional case.
  - 4. The case of the simplest diffeomorphism in Rn.
  - 5. Composition of mappings and formula for changing variables.
  - 6. Additivity of the integral and completion of the proof of the formula for changing variables in the integral.
  - 7. Some consequences and generalizations of the formula for changing variables in multiple integrals.
- § 6. Improper multiple integrals
  - 1. Basic definitions.
  - 2. Majorant gain for the convergence of an improper integral.
  - 3. Change of variables in the improper integral.
Chapter XII. Surfaces and differential forms in Rn
- § 1. Surfaces in Rn
- § 2. Surface orientation
- § 3. The edge of the surface and its orientation
  - 1. Surface with an edge.
  - 2. Coordination of surface and edge orientation.
- § 4. Surface area in Euclidean space
- § 5. Initial information about differential forms
  - 1. Differential form, definition and examples.
  - 2. Coordinate notation of differential form.
  - 3. External differential shape.
  - 4. Transfer of vectors and shapes during mappings.
  - 5. Shapes on surfaces.
Chapter XIII. Curvilinear and surface integrals
- § 1. Integral of a differential form
  - 1. Initial tasks, guiding considerations, examples.
  - 2. Determination of the integral of shape over an oriented surface.
- § 2. Volume form, integrals of the first and second kind
  - 1. Mass of the material surface.
  - 2. Surface area as an integral of the shape.
  - 3. Volume shape.
  - 4. Expression of the volume shape in Cartesian coordinates.
  - 5. Integrals of the first and second kind.
- § 3. Basic integral formulas of analysis
  - 1. Green's formula.
  - 2. Gauss-Ostrogradsky formula.
  - 3. Stokes formula in R3.
  - 4. General Stokes formula.
Chapter XIV. Elements of vector analysis and field theory
- § 1. Differential operations of vector analysis
  - 1. Scalar and vector fields
  - 2. Vector fields and forms in R3.
  - 3. Differential operators grad, rot, div and V.
  - 4. Some differential formulas of vector analysis.
  - 5. Vector operations in curvilinear coordinates.
- § 2. Integral formulas of field theory
  - 1. Classical integral formulas in vector notation.
  - 2. Physical interpretation.
  - 3. Some further integral formulas.
- § 3. Potential fields
  - 1. Vector field potential.
  - 2. Necessary condition of potentiality.
  - 3. Potentiality criterion for a vector field.
  - 4. Topological structure of the region and potential.
  - 5. Vector potential. Precise and closed forms.
- § 4. Application examples
  - 1. Thermal conductivity equation.
  - 2. Continuity equation.
  - 3. Basic equations of continuum dynamics.
  - 4. Wave equation.
Chapter XV. Integration of differential forms on manifolds 305
- § 1. Some reminders from linear algebra
  - 1. Algebra of forms.
  - 2. Algebra of skew-symmetric forms.
  - 3. Linear mappings of linear spaces, and conjugate mappings of conjugate spaces. Tasks and exercises
- § 2. Diversity.
  - 1. Definition of diversity.
  - 2. Smooth manifolds and smooth mappings.
  - 3. Orientation of the manifold and its edges.
  - 4. Partitioning of unity and realization of manifolds in the form of surfaces in Rn.
- § 3. Differential forms and their integration on manifolds
  - 1. Tangent space to a manifold at a point.
  - 2. Differential form on a manifold.
  - 3. External differential.
  - 4. Integral of form over a manifold.
  - 5. Stokes formula.
- § 4. Closed and exact forms on a manifold
  - 1. Poincare's theorem.
  - 2. Homology and cohomology.
Chapter XVI. Uniform convergence and basic operations of analysis over series and families of functions
- § 1. Pointwise and uniform convergence
  - 1. Pointwise convergence.
  - 2. Statement of basic questions.
  - 3. Convergence and uniform convergence of a family of functions depending on a parameter.
  - 4. Cauchy criterion for uniform convergence.
- § 2. Uniform convergence of series of functions
  - 1. Basic definitions and criterion for uniform convergence of a series.
  - 2. Weyergatrass test for uniform convergence of a series.
  - 3. Abel-Dirichlet test.
- § 3. Functional properties of the limit function
  - 1. Specification of the task.
  - 2. Conditions for commutation of two limit passages.
  - 3. Continuity and passage to the limit.
  - 4. Integration and passage to the limit.
  - 5. Differentiation and passage to the limit.
- § 4. Compact and dense subsets of the space of continuous functions
  - 1. The Arcela-Ascoli theorem.
  - 2. Metric space.
  - 3. Stone's theorem.
Chapter XVII. Integrals depending on a parameter
- § 1. Proper integrals depending on the parameter
  - 1. The concept of an integral depending on a parameter.
  - 2. Continuity of the integral depending on the parameter.
  - 3. Differentiation of an integral depending on a parameter.
  - 4. Integration of an integral depending on a parameter
- § 2. Improper integrals depending on a parameter
  - 1. Uniform convergence of an improper integral with respect to a parameter.
  - 2. Passage to the limit under the sign of an improper integral and continuity of an improper integral depending on a parameter.
  - 3. Differentiation of an improper integral with respect to a parameter.
  - 4. Integration of an improper integral over a parameter.
- § 3. Euler integrals
  - 1. Beta function.
  - 2. Gamma function.
  - 3. Relationship between functions B and D.
  - 4. Some examples.
- § 4. Convolution of functions and initial information about generalized functions
  - 1. Convolution into physical problems(guiding thoughts).
  - 2. Some general properties of convolution.
  - 3. Delta-shaped families of functions and Weierstrass’s approximation theorem.
  - 4. Initial ideas about distributions.
- § 5. Multiple integrals depending on a parameter
  - 1. Proper multiple integrals depending on the parameter.
  - 2. Improper multiple integrals depending on the parameter.
  - 3. Improper integrals with variable singularity.
  - 4. Convolution, fundamental solution and generalized functions in the multidimensional case.
Chapter XVIII Reed Fourier and the Fourier Transform
- § 1. Basic general concepts related to the concept of Fourier series
  - 1. Orthogonal systems of functions.
  - 2. Fourier coefficients and Fourier series.
  - 3. About one important source of orthogonal systems of functions in analysis.
- § 2. Trigonometric Fourier series
  - 1. Main types of convergence of the classical Fourier series.
  - 2. Study of the pointwise convergence of the trigonometric Fourier series.
  - 3. Smoothness of the function and the rate of decrease of the Fourier coefficients.
  - 4. Completeness of the trigonometric system.
- § 3. Fourier transform
  - 1. Representation of a function by the Fourier integral.
  - 2. Regularity of the function and the rate of decrease of its Fourier transform.
  - 3. The most important hardware properties of the Fourier transform.
  - 4. Application examples.
Chapter XIX. Asymptotic expansions
- § 1. Asymptotic formula and asymptotic series
  - 1. Basic definitions.
  - 2. General information about asymptotic series.
  - 3. Power asymptotic series.
- § 2. Asymptotic behavior of integrals (Laplace’s method)
  - 1. The idea of Laplace's method.
  - 2. The principle of localization of the length of the Laplace integral.
  - 3. Canonical integrals and their asymptotics.
  - 4. The main term of the asymptotics of the Laplace integral.
  - 5. Asymptotic expansions of Laplace integrals.
Tasks and exercises
Literature
Index of basic symbols
Alphabetical index

Transcript

2 Mathematical analysis 1. Completeness: supremum and infimum of a number set. The principle of nested segments. Irrationality of number Theorem on the existence of the limit of a monotone sequence. Number e. 3. Equivalence of definitions of the limit of a function at a point in the language and in the language of sequences. Two great limits. 4. Continuity of a function of one variable at a point, discontinuity points and their classification. Properties of a function continuous on an interval. 5. Weierstrass's theorems on the greatest and least values of a continuous function defined on a segment. 6. Uniformity of continuity. Cantor's theorem. 7. The concept of derivative and differentiability of a function of one variable, differentiation of a complex function. 8. Derivatives and differentials of higher orders of a function of one variable. 9. Study of a function using derivatives (monotonicity, extrema, convexity and inflection points, asymptotes). 10. Parametrically defined functions and their differentiation. 11. Theorems of Rolle, Lagrange and Cauchy. 12. L'Hopital's rule. 13. Taylor's formula with a remainder term in Lagrange form. 14. Local Taylor formula with remainder term in Peano form. Expansion of basic elementary functions using the Taylor formula. 15. Riemann criterion for integrability of a function. Classes of integrable functions. 16. Theorem on the existence of an antiderivative for every continuous function. Newton-Leibniz formula. 17. Integration by parts and change of variable in the indefinite integral. Integrating rational fractions. 18. Methods for approximate calculation of definite integrals: methods of rectangles, trapezoids, parabolas. 19. Definite integral with variable upper limit; mean value theorems. 20. Geometric applications of the definite integral: area of a flat figure, volume of a body in space. 21. Power series; expansion of functions into power series. 22. Improper integrals of the first and second kind. Signs of convergence. 23. The simplest conditions for uniform convergence and term-by-term differentiation of the trigonometric Fourier series. 24. Sufficient conditions for differentiability at a point of a function of several variables. 25. Definition, existence, continuity and differentiability of an implicit function. 26. Necessary condition for a conditional extremum. Lagrange multiplier method. 27. Number series. Cauchy criterion for series convergence. 28. Cauchy's test for the convergence of positive series 29. D'Alembert's test for the convergence of positive series 30. Leibniz's theorem on the convergence of an alternating series. 31. Cauchy criterion for uniform convergence of functional series. 32. Sufficient conditions for continuity, integrability and differentiability of the sum of a functional series. 33. Structure of the convergence set of an arbitrary functional series. The Cauchy-Hadamard formula and the structure of the convergence set of a power series.

3 34. Multiple Riemann integral, its existence. 35. Reducing a multiple integral to a repeated one. References 1. Kartashev, A.P. Mathematical analysis: textbook. - 2nd ed., stereotype. - St. Petersburg: Lan, p. 2. Kirkinsky, A.S. Mathematical analysis: textbook for universities. - M.: Academic Project, p. 3. Kudryavtsev, L.D. Short course in mathematical analysis. Vol. 1, 2. Differential and integral calculus of functions of many variables. Harmonic analysis: a textbook for university students.- Ed. 3rd, revised - Moscow: Fizmatlit, p. 4. Mathematical analysis. T. 1.2: / ed. V.A. Sadovnichego. - M.: Scientific Research Center "RHD", Nikolsky, S.M. Course of mathematical analysis. T. 1, 2.- Ed. 4th, revised and additional - Moscow: Science, p. 6. Ilyin, V.A. Fundamentals of mathematical analysis. Part 1, 2. - Ed. 4th, revised and additional - Moscow: Science, p. Differential equations. 1. The theorem on the existence and uniqueness of a solution to the Cauchy problem for a first-order ordinary differential equation. 2. The theorem on the existence and uniqueness of the solution to the Cauchy problem for a first-order ordinary differential equation. 3. The theorem on the continuous dependence of the solution to the Cauchy problem for a first-order ordinary differential equation on the parameters and on the initial data. 4. The theorem on the differentiability of the solution to the Cauchy problem for a first-order ordinary differential equation with respect to parameters and initial data. 5. Linear ordinary differential equations (ODE). General properties. Homogeneous ODE. Fundamental system of solutions. Vronskian. Liouville formula. General solution of a homogeneous ODE. 6. Inhomogeneous linear ordinary differential equations. Common decision. Lagrange's method of variation of constants. 7. Homogeneous linear ordinary differential equations with constant coefficients. Construction of a fundamental system of solutions. 8. Inhomogeneous linear ordinary differential equations with constant coefficients with inhomogeneity in the form of a quasi-polynomial (non-resonant and resonant cases). 9. Homogeneous system of linear ordinary differential equations (ODE). Fundamental system of solutions and fundamental matrix. Vronskian. Liouville formula. Structure general solution homogeneous system ODU. 10. Inhomogeneous system of linear ordinary differential equations. Lagrange's method of variation of constants. 11. Homogeneous system of linear differential equations with constant coefficients. Construction of a fundamental system of solutions. 12. Inhomogeneous system of ordinary differential equations with constant coefficients with inhomogeneity in the form of a matrix with elements of quasi-polynomials (non-resonant and resonant cases). 13. Statement of boundary value problems for a second order linear ordinary differential equation. Special functions of boundary value problems and their explicit representations. Green's function and its explicit representations. Integral representation

4 solutions to the boundary value problem. Theorem of existence and uniqueness of a solution to a boundary value problem. 14. Autonomous systems. Properties of solutions. Singular points of a linear autonomous system of two equations. Stability and asymptotic stability according to Lyapunov. Stability of a homogeneous system of linear differential equations with a variable matrix. 15. First approximation stability of a system of nonlinear differential equations. Second Lyapunov method. References 1. Samoilenko, A.M. Differential equations: practical course: textbook for university students.- Ed. 3rd, revised - Moscow: Higher School, p. 2. Agafonov, S.A. Differential equations: textbook. - 4th ed., revised - M.: Publishing House of MSTU named after N.E. Bauman, p. 3. Egorov, A.I. Ordinary differential equations with applications - Ed. 2nd, corrected - Moscow: FIZMATLIT, p. 4. Pontryagin, L.S. Ordinary differential equations.- Ed. 6th - Moscow; Izhevsk: Regular and chaotic dynamics, p. 5. Tikhonov, A.N. Differential equations: a textbook for students of physical specialties and the specialty "Applied Mathematics". - Ed. 4th, ster. - Moscow: Fizmatlit, p. 6. Phillips, G. Differential equations: translation from English / G. Phillips; edited by A.Ya. Khinchin. - 4th ed., printed. - Moscow: KomKniga, p. Algebra and number theory 1. Definition of group, ring and field. Examples. Construction of a field of complex numbers. Raising complex numbers to powers. Extracting roots from complex numbers. 2. Matrix algebra. Types of matrices. Operations on matrices and their properties. 3. Determinants of matrices. Definition and basic properties of determinants. Inverse matrices. 4. Systems of linear algebraic equations (SLAEs). Study of SLAU. Gauss method. Cramer's rule. 5. Ring of polynomials in one variable. Theorem on division with remainder. GCD of two polynomials. 6. Roots and multiple roots of a polynomial. Fundamental theorem of algebra (without proof). 7. Linear spaces. Examples. Basis and dimension of linear spaces. Matrix of transition from one basis to the second basis. 8. Subspaces. Operations on subspaces. Direct sum of subspaces. Criteria for direct sum of subspaces. 9. Matrix rank. Compatibility of SLAU. Kronecker-Capelli theorem. 10. Euclidean and unitary spaces. Metric concepts in Euclidean and unitary spaces. Cauchy-Bunyakovsky inequality. 11. Orthogonal vector systems. Orthogonalization process. Orthonormal bases. 12. Subspaces of unitary and Euclidean spaces. Orthogonal addition. 13. Linear operators in linear spaces and operations on them. Linear operator matrix. Matrices of a linear operator in various bases.

5 14. Image and kernel, rank and defect of a linear operator. Dimensions of the kernel and image. 15. Invariant subspaces of a linear operator. Eigenvectors and eigenvalues of the linear operator. 16. Diagonalizability criterion for a linear operator. Hamilton-Cayley theorem. 17. Jordan basis and Jordan normal form of the matrix of a linear operator. 18. Linear operators in Euclidean and unitary spaces. Conjugate, normal operators and their simple properties. 19. Quadratic forms. Canonical and normal form of quadratic forms. 20. Constant sign quadratic forms, Sylvester criterion. 21. Divisibility relation in the ring of integers. Theorem on division with remainder. GCD and LCM of integers. 22. Continued (continued) fractions. Matching fractions. 23. Prime numbers. Sieve of Eratosthenes. Infinity theorem prime numbers. Factoring a number into prime factors 24. Antje function. Multiplicative function. Möbius function. Euler function. 25. Comparisons. Basic properties. Complete system deductions. The given system of deductions. Euler's and Fermat's theorems. 26. Comparisons of the first degree with one unknown. System of comparisons of the first degree. Chinese remainder theorem. 27. Comparisons of any degree by composite module. 28. Comparisons of the second degree. Legendre's symbol. 29. Primordial roots. 30. Indexes. Applying indexes to solving comparisons. References 1. Kurosh, A.G. Lectures on general algebra: textbook / A.G. Kurosh. - 2nd ed., printed - St. Petersburg: Publishing House "Lan", p. 2. Birkhoff, G. Modern applied algebra: a textbook / Garrett Birkhoff, Thomas K. Barty; translation from English by Yu.I. Manina. - 2nd ed., printed. - St. Petersburg: Lan, p. 3. Ilyin, V.A. Linear algebra: a textbook for students of physical specialties and the specialty "Applied Mathematics". - Ed. 5th, ster. - Moscow: FIZMATLIT, Kostrikin, A.I. Introduction to algebra. Part 1. Fundamentals of Algebra: a textbook for university students studying in the specialties "Mathematics" and "Applied Mathematics". - Ed. 2nd, rev. - Moscow: FIZMATLIT, Vinogradov, I.M. Fundamentals of number theory: textbook.- Ed. 11th - St. Petersburg; Moscow; Krasnodar: Lan, s. 6. Bukhshtab, A.A. Number theory: textbook. - 3rd ed., stereotype. - St. Petersburg; Moscow; Krasnodar: Lan, s. Geometry 1. Scalar, vector and mixed products of vectors and their properties. 2. Equation of a straight line on a plane defined in various ways. The relative position of two straight lines. The angle between two straight lines. 3. Transformation of coordinates when moving from one Cartesian coordinate system to another. 4. Polar, cylindrical and spherical coordinates. 5. Ellipse, hyperbola and parabola and their properties. 6. Classification of second order lines.

6 7. Equation of a plane defined in various ways. The relative position of two planes. Distance from a point to a plane. The angle between two planes. 8. Equations of a straight line in space. The relative position of two lines, a straight line and a plane. Distance from a point to a line. The angle between two straight lines, a straight line and a plane. 9. Ellipsoids, hyperboloids and paraboloids. Rectilinear generators of second-order surfaces. 10. Surfaces of rotation. Cylindrical and conical surfaces. 11. Definition of an elementary curve. Methods for defining a curve. Curve length (definition and calculation). 12. Curvature and torsion of a curve. 13. Accompanying benchmark of a smooth curve. Frenet's formulas. 14. The first quadratic form of a smooth surface and its applications. 15. Second quadratic form of a smooth surface, normal curvature of the surface. 16. Main directions and main curvatures of the surface. 17. Lines of curvature and asymptotic lines of the surface. 18. Average and Gaussian surface curvature. 19. Topological space. Continuous displays. Homeomorphisms. Examples. 20. Euler characteristic of a manifold. Examples. Literature 1. Nemchenko, K.E. Analytical geometry: textbook. - Moscow: Eksmo, p. 2. Dubrovin, B.A. Modern geometry: methods and applications. Vol. 1, 2. Geometry and topology of manifolds. - 5th ed. rev. - Moscow: Editorial URSS, p. 3. Zhafyarov, A.Zh. Geometry. In 2 hours, textbook. - 2nd ed. - Novosibirsk: Siberian University Publishing House, p. 4. Efimov, N.V. A short course in analytical geometry: a textbook for students of higher educational institutions. - 13th ed. - Moscow: FIZMATLIT, p. 5. Taimanov, I.A. Lectures on differential geometry. - Moscow; Izhevsk: Institute of Computer Research, p. 6. Atanasyan L.S., Bazyrev V.T. Geometry, part 1,2. Moscow: Knorus, p. 7. Rashefsky P.S. Course of differential geometry. Moscow: Nauka, p. Theory and methods of teaching mathematics 1. Contents of teaching mathematics in high school. 2. Didactic principles of teaching mathematics. 3. Methods of scientific knowledge. 4. Visualization when teaching mathematics. 5. Forms, methods and means of monitoring and assessing students’ knowledge and skills. Marking standards. 6. Extracurricular work in mathematics. 7. Mathematical concepts and methods of their formation. 8. Problems as a means of teaching mathematics. 9. In-depth study of mathematics: content, techniques and forms of organization of training. 10. Types of mathematical propositions: axiom, postulate, theorem.

7 11. Math lesson notes. 12. Math lesson. Types of lessons. Lesson analysis. 13. Studying mathematics in a small school: content, techniques and forms of teaching organization. 14. New teaching technologies. 15. Differentiation of teaching mathematics. 16. Individualization of teaching mathematics. 17. Motivation educational activities schoolchildren. 18. Logical-didactic analysis of the topic. 19. Technological approach to teaching mathematics 20. Humanization and humanization of teaching mathematics. 21. Education in the process of teaching mathematics. 22. Methodology for studying identity transformations. 23. Methodology for studying inequalities. 24. Methodology for studying function. 25. Methodology for studying the topic “Equations and inequalities with modulus.” 26. Methodology for studying the topic “Cartesian coordinates”. 27. Methods for studying polyhedra and round bodies. 28. Methodology for studying the topic “Vectors”. 29. Methodology for solving motion problems. 30. Methodology for solving problems for joint work. 31. Methodology for studying the topic “Triangles” 32. Methodology for studying the topic “Circle and Circle”. 33. Methodology for solving problems on alloys and mixtures. 34. Methodology for studying the topic “Derivative and integral”. 35. Methodology for studying the topic “ Irrational equations and inequality." 36. Methodology for studying the topic “Solving equations and inequalities with parameters.” 37. Methodology for studying the basic concepts of trigonometry. 38. Methodology for studying the topic “Trigonometric equations” 39. Methodology for studying the topic “Trigonometric inequalities”. 40. Methodology for studying the topic “Inverse trigonometric functions.” 41. Methodology for studying the topic “General methods for solving equations in a school mathematics course.” 42. Methodology for studying the topic “Quadratic equations”. 43. Methodology for studying the basic concepts of stereometry 44. Methodology for studying the topic “ Common fractions" 45. Methodology for studying the topic “Use of derivatives in the study of functions” References 1. Argunov, B.I. School course mathematics and methods of teaching it. - Moscow: Education, p. 2. Zemlyakov, A.N. Geometry in 11th grade: methodological recommendations for studies. A.V. Pogorelova: a manual for teachers. - 3rd ed., back. - M.: Education, p. 3. Studying algebra in grades 7-9: a book for teachers / Yu.M. Kolyagin, Yu.V. Sidorov, M.V. Tkacheva and others - 2nd ed. - M.: Prosveshchenie, p. 4. Latyshev, L.K. Translation: theory, practice and teaching methods: textbook. - 3rd ed., ster. - Moscow: Academy, p. 5. Methods and technology of teaching mathematics: a course of lectures: a textbook for students of mathematical faculties of higher educational institutions studying in the direction (050200) physics and mathematics education. - Moscow: Bustard, p.

8 6. Roganovsky, N.M. Methods of teaching mathematics in secondary school: textbook. - Minsk: Higher School, p.

25. Definition, existence, continuity and differentiability of an implicit function. 26. Necessary condition for a conditional extremum. Lagrange multiplier method. 27. Number series. Cauchy convergence criterion

Ministry of Education and Science Russian Federation Federal state budget educational institution higher vocational education"SIBERIAN STATE GEODETIC ACADEMY"

Ministry of Education and Science of the Republic of Kazakhstan RSE RSE "Eurasian National University them. L.N. Gumilyov" Department of Fundamental Mathematics PROGRAM for the entrance exam to doctoral studies

MINISTRY OF EDUCATION AND SCIENCE OF RUSSIA Federal state budgetary educational institution higher education"Chelyabinsk State University" (FSBEI HE "ChelSU") APPROVED BY: Chairman admissions committee,

EAST KAZAKHSTAN STATE TECHNICAL UNIVERSITY NAMED AFTER D. SERIKBAEVA Faculty of Information Technologies and Business APPROVED by Dean of FITB N. Denisova 2016 ENTRANCE EXAMINATION PROGRAM

1. The purpose of studying the discipline is: to prepare a highly professional specialist with mathematical knowledge, skills and abilities to use mathematics as a tool for logical analysis, numerical

Ministry of Education and Science of the Russian Federation Federal State Budgetary Educational Institution of Higher Professional Education "Ivanovo State University" Faculty of Mathematics and Computer Science PROGRAM FOR THE ENTRANCE TEST TO THE MASTER'S PROGRAM for studying in

EAST KAZAKHSTAN STATE TECHNICAL UNIVERSITY NAMED AFTER D. SERIKBAEVA Faculty of Information Technologies and Business APPROVED Dean of FITiB N. Denisova 2016 ENTRANCE EXAMINATION PROGRAM

Abstract to the work program of the discipline Author Fedorov Yu.I., Associate Professor Name of the discipline: B1.B.05 Mathematics Purpose of mastering the discipline: - formation of knowledge, skills, skills in mathematics necessary

CONTENTS PART I Lectures 1 2 Determinants and matrices Lecture 1 1.1. The concept of a matrix. Types of matrices... 19 1.1.1. Basic definitions... 19 1.1.2. Types of matrices... 19 1.2.* Permutations and substitutions... 21 1.3.*

List of exam questions: 1st semester 1. Sets and operations on them. 2. Cartesian product of sets. 3. Limit points. 4. Sequence limit. 5. Function limit. 6. Infinitesimal.

“APPROVED” Acting Director of FMITI Pop E. N. 2018 PROGRAM of the entrance exam to the master's program in the direction of 01.04.01. MATHEMATICS, master's program "Complex Analysis" Entrance program

Methodological materials for teachers. Approximate plans lecture classes. Section “Algebra: basic algebraic structures, linear spaces and linear mappings” Lecture 1 on the topic “Complex

Preface Chapter I. ELEMENTS OF LINEAR ALGEBRA 1. Matrices 1.1. Basic concepts 1.2. Operations on matrices 2. Determinants 2.1. Basic concepts 2.2. Properties of determinants 3. Non-singular matrices 3.1.

Preface Chapter I. ELEMENTS OF LINEAR ALGEBRA 1. Matrices 1.1. Basic concepts 1.2. Actions on matrices 2. Determinants 2.1. Basic concepts 2.2. Properties of determinants 3. Non-singular matrices 3.1.

APPROVED by the manager. Department of Physics and Mathematics Disciplines E.N. Kiryukhova 20, protocol Questions for the exam in the discipline “Mathematics” Specialties “ Information Systems and technology" by correspondence

This course of lectures is intended for all categories of university students studying higher mathematics to one degree or another. The first part contains required material in 9 sections of the higher mathematics course,

4. Abstract to the work program of the discipline Author Yu.I. Fedorov, Associate Professor Name of the discipline: B1.B.04 Higher Mathematics Purpose of mastering the discipline: - formation of knowledge, skills, proficiency in higher mathematics

1. The purpose and objectives of the discipline Mathematical Analysis The purpose of mastering the discipline “Mathematical Analysis” is to develop in future specialists the knowledge and ability to use mathematical apparatus and mathematical

Ministry of Science and Higher Education of the Russian Federation Federal State Budgetary Educational Institution of Higher Education “Kaluga State University named after. K.E. Tsiolkovsky"

NAN CHOU VO Academy of Marketing and Social Information Technologies ABSTRACT OF THE ACADEMIC DISCIPLINE Direction of training 10.03.01 “Information Security” focus (profile) of the program Organization

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Professional Education "SAMARA STATE UNIVERSITY" Mechanics and Mathematics

CONTENTS Preface... 15 Chapter I. ELEMENTS OF LINEAR ALGEBRA 1. Matrices... 16 1.1. Basic concepts... 16 1.2. Actions on matrices... 17 2. Determinants... 20 2.1. Basic concepts... 20 2.2. Properties

EAST KAZAKHSTAN STATE TECHNICAL UNIVERSITY NAMED AFTER D. SERIKBAEVA Faculty of Information Technologies and Energy APPROVED by Vice-Rector for Academic and Methodological Work Link N.N. 2014

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal state budgetary educational institution of higher professional education "UFA STATE AVIATION TECHNICAL

Questions of the entrance exam to the master's program in the specialty “6M070500-Mathematical and computer modeling” Mathematical analysis I, II, III 1. Completeness: the existence of a limit of a monotonic sequence.

FEDERAL AGENCY FOR EDUCATION STATE EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION "TYUMEN STATE OIL AND GAS UNIVERSITY" INSTITUTE OF CYBERNETICS, INFORMATION SCIENCE

FEDERAL EDUCATION AGENCY State educational institution of higher professional education “Ural State University named after. A.M. Gorky" Mathematics and Mechanics

CONTENTS Preface 3 Introduction 5 Part one. Mathematical analysis of functions of one variable 10 Chapter I. Real numbers 10 1. Sets. Notation. Logical symbols 10 2. Real numbers

Ministry of Education and Science of the Krasnodar Territory state budgetary professional educational institution of the Krasnodar Territory "Krasnodar Information Technology College" lesson

Ministry of Education and Science of the Russian Federation Federal State Budgetary Educational Institution of Higher Professional Education "Yaroslavl State Pedagogical University them. K.D. Ushinsky" U T V E R J D A Y First Vice-Rector M.V. Novikov 20 PROGRAM

Program of a comprehensive exam in the specialty 6M060100-Mathematics Tickets for the entrance exam to the master's program in the specialty 6M060100 "Mathematics" are compiled for the main mathematical disciplines

REAL AND COMPLEX ANALYSIS 1. Mathematical analysis Theory of limits. Series theory. Basic theorems about continuous functions. Basic theorems of differential calculus. (theorem about average values,

Appendix 3 MINISTRY OF SCIENCE AND EDUCATION OF THE RUSSIAN FEDERATION Federal State Autonomous Educational Institution of Higher Professional Education "Kazan (Volga Region) Federal University" APPROVED by Vice-Rector R.G. Minzaripov 20 MP RECOMMENDED by the Decision of the Scientist

Department of Mathematical Analysis and Theory of Functions Schedule training sessions in the discipline mathematical analysis Specialty index SF course I semester 1 Leading the discipline Ph.D., associate professor Budochkina

Abstract to the work program of discipline B1.B.4 Mathematics Direction of training Training profile 03/05/01 Geology Geophysics Graduate qualification (degree) Bachelor Form of study full-time Course 1,

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION FEDERAL STATE AUTONOMOUS EDUCATIONAL INSTITUTION OF HIGHER EDUCATION "NOVOSIBIRSK NATIONAL RESEARCH STATE

(3) MATHEMATICAL ANALYSIS Department of Higher Mathematics MMF Author of the program: Associate Professor M.P. Vishnevsky Lecturer: 1st semester 1. Introduction. Sets and operations on them. Set mappings. Countable sets. Valid

Program of the entrance exam to the master's program in the specialty “6M060100-MATHEMATICS” Mathematical analysis Numerical function and methods for specifying it. Limit of a function and main theorems, definitions. Criteria

ENTRANCE TEST PROGRAM for the educational program of higher education, the program for training scientific and pedagogical personnel in graduate school of the Federal State Budgetary Educational Institution of Higher Education "Oryol State University named after

QUESTIONS AND MODEL PROBLEMS for the final exam in the discipline “Mathematical Analysis” Applied Mathematics In the oral exam, the student receives two theoretical issues and two problems Total 66 questions per year

Abstract of the work program of the discipline Mathematical analysis (name of the discipline) Direction of training 03.03.02 physics Profile of training “Fundamental physics”, “Physics” atomic nucleus and particles"

FEDERAL STATE EDUCATIONAL BUDGETARY INSTITUTION OF HIGHER EDUCATION FINANCIAL UNIVERSITY UNDER THE GOVERNMENT OF THE RUSSIAN FEDERATION (Penza branch) Department of Management, Informatics and

Course program "Mathematical Analysis". Semester 1 (72 hours of lectures, 72 hours of practical classes) Thematic plan of lectures. I. Introduction to analysis. 1. Elements of set theory. 2. Natural numbers. Mathematical

QUESTIONS for the final exam 7/8 in the discipline “Mathematical Analysis” Program “Applied Mathematics” At the oral exam, the student receives two theoretical questions and two problems.. What is numerical

Matrix. Algebra and geometry 1. Determinants. Decomposition of the determinant in row and column. Algebra 2. Geometric vectors. Dot product of vectors. Vector and mixed work vectors.

Approved at the meeting of the Department of Mathematics and Informatics, Minutes 2(25) September 8, 2015. head Department of Ph.D. Timshina D.V. Questions for testing in the discipline “LINEAR ALGEBRA AND MATHEMATICAL ANALYSIS”

Funds Funds of assessment tools in discipline B.2.1 “Mathematical analysis” for ongoing monitoring of progress and intermediate certification students in the direction 080100.62 “Economics” Topics

2 Intermediate certification tests in the discipline: List of questions for testing in the discipline “Mathematics” I semester I Elements of linear algebra 1. The concept of 2nd and 3rd order determinants, their calculation and

MINORSKY V. P. Collection of problems in higher mathematics: Textbook. allowance for colleges. 13th ed. M.: Publishing House of Physico-Mathematical Literature, 2010. 336 with ISBN 9785-94052-184-6. CONTENTS FROM THE AUTHOR'S FOREWORD

1 2 1. GOALS AND OBJECTIVES OF PRACTICAL LESSONS Practical classes in the discipline “Mathematics” are conducted with the aim of: 1. Formation of skills: - systematize the knowledge gained in lecture classes and practical

State Committee RSFSR for Science and Higher Education SIBERIAN STATE GEODETIC ACADEMY V.P. Verbnaya D.A. KRIMSKIKH E.S. PLYUSNINA HIGHER MATHEMATICS Methodological manual for students

GBOU SPO Prokopyevsk Polytechnic College PROGRAM OF THE ACADEMIC DISCIPLINE “ELEMENTS OF HIGHER MATHEMATICS” Recommended for specialty 30111 Computer networks Name of basic training qualification

SHORT PROGRAM OF ENTRANCE EXAMINATIONS TO THE MASTER'S PROGRAM IN MATHEMATICS EDUCATION 2015 Section 1. Algebra and number theory 1. Algebraic and trigonometric forms of complex numbers.

Program of the written exam in “Higher Mathematics” for the first year correspondence departments Faculty of Economics in the winter session The written exam is held for two hours. On the exam, every student

ASSESSMENT TOOLS FOR CURRENT ACHIEVEMENT CONTROL AND INTERMEDIATE CERTIFICATION BASED ON THE RESULTS OF MASTERING THE DISCIPLINE Academic discipline B.2.1 - Mathematics Preparation profile: Production management Topics

Federal Agency for Education State Educational Institution of Higher Professional Education "Pomeranian State University named after M.V. Lomonosov" APPROVED by the Rector of Pomorsky state university named after M.V. Lomonosova I.R. Lugovskaya

QUESTIONS FOR PREPARING FOR THE EXAM Vector algebra and analytical geometry. Definition of a vector. Equality of vectors. Linear operations on vectors. Linear dependence of vectors. Basis and coordinates.

2 Intermediate certification tests in the discipline: List of questions for exams in the discipline “Mathematics” I Elements of linear algebra I semester 1. Determinants. Properties of determinants. 2. Matrices. Kinds