Mathematical analysis. Mathematical analysis Mathematical analysis pdf

Books. Download DJVU books, PDF for free. Free electronic library
V.A. Zorich, Mathematical Analysis (Part 2)

You can (the program will mark it in yellow)
You can see the list of books on higher mathematics sorted alphabetically.
You can see the list of books on higher physics sorted alphabetically.

Ladies and gentlemen!! In order to download files of electronic publications without "glitches", click on the underlined link with the file RIGHT mouse button, select a command "Save target as ..." ("Save target as...") and save the e-pub file to your local computer. Electronic publications are typically in Adobe PDF and DJVU formats.

Chapter IX. Continuous displays ( general theory)

§ 1. Metric space
1. Definitions and examples (11).
2. Open and closed subsets of a metric space (13).
3. Subspace of the metric space (17).
4. Direct product of metric spaces (18).

§ 2. Topological space
1. Basic definitions (19).
2. Subspace of the topological space (23).
3. Direct product of topological spaces. (24).

§ 3. Compacta
1. Definition and general properties compacta (25).
2. Metric compacts (27).

§ 4. Radiant topological spaces

§ 5. Complete metric spaces K Basic definitions and examples (31).
2. Completion of the metric space (34).

§ 6. Continuous mappings of topological spaces
1. Display limit (38).
2. Continuous mappings (40).

§ 7. The principle of contraction mappings

Chapter X. Calculus from a more general point of view

§ 1. Linear normed space
1. Some examples of linear spaces of analysis (50).
2. Norm in vector space (51).
3. Scalar product in vector space (54).

§ 2. Linear and multilinear operators 67
1. Definitions and examples (57).
2. The norm of the operator (64)).
3. Space of continuous operators (64).

§ 3. Mapping differential
1. Mapping differentiable at a point (69).
2. General laws differentiation (70).
3. Some examples (71).
4. Partial derivatives of mappings (77).

§ 4. Finite increment theorem and some examples of its use
1. Finite increment theorem (80)
2. Some examples of the application of the finite increment theorem (83).

§ 5. Derivative mappings of higher orders
1. Definition of the nth differential (87).
2. Derivative with respect to the vector and calculation of the values ​​of the nth differential (88).
3. Symmetry of differentials higher order (89).
4. Some remarks (91).

§ 6. The Taylor formula and the study of extrema
1. Taylor formula for mappings (93).
2. Study of internal extremes (94).
3. Some examples (96).

§ 7. General implicit function theorem

Chapter XI. Multiple integrals 115

§ 1. The Riemann integral on an n-dimensional interval
1. Definition of the integral (113).
2. Lebesgue criterion for the integrability of a function in the sense of Pnman (115).
3. Darboux criterion (120).

§ 2. Integral over a set
1. Admissible sets (123).
2. Integral over a set (124)
3. Measure (volume) of an admissible set (125).

§ 3. General properties of the integral
1. Integral as a linear functional (127).
2. Additivity of the integral (127).
3. Estimates of the integral (128).

§ 4. Reduction of a multiple integral to an iterated one
1. Fubini's theorem (131).
2. Some consequences (134).

§ 5. Change of variables in a multiple integral 139
1. Statement of the question and heuristic derivation of the formula - change of variables (139).
2. Measurable sets and smooth mappings (141).
3. One-dimensional case (143).
4. The case of the simplest diffeomorphism in Rn (145).
5. Composition of mappings and formula for change of variables (146).
6. Additivity of the integral and completion of the proof of the formula for the change of variables in the integral (147).
7. Some consequences and generalizations of the formula for the change of variables in multiple integrals (148).

§ 6. Improper multiple integrals
1. Basic definitions (154).
2. Majorant Approach to the Convergence of the Improper Integral (157).
3. Change of variables in the improper integral (159).

Chapter XII. Surfaces and differential forms in Rn

§ 1. Surfaces in Rn

§ 2. Surface orientation

§ 3. Surface edge and its orientation
1. Surface with edge (182).
2. Coordination of surface and edge orientation (184).

§ 4. Surface area in Euclidean space

§ 5. Introduction to differential forms
1. Differential form, definition and examples (197).
2. Coordinate notation of the differential form (200).
3. External differential of the form (203).
4. Transfer of vectors and shapes in mappings (206).
5. Forms on surfaces (209).

Chapter XIII. Curvilinear and surface integrals

§ 1. Integral of differential form
1. Initial problems, suggestive considerations, examples (213).
2. Definition of the integral of the shape over an oriented surface (219).

§ 2. Volume form, integrals of the first and second kind
1. Mass of the material surface (227).
2. Area of ​​the surface as an integral of the form (228).
3. Volume shape (229).
4. Expression of the shape of the volume in Cartesian coordinates (231).
5. Integrals of the first and second kind (232).

§ 3. Basic integral formulas of analysis
1. Green's formula (236).
2. Gauss-Ostrogradsky formula (241).
3. Stokes formula in R3 (244).
4. General formula Stokes (246).

Chapter XIV. Elements of vector analysis and field theory

§ 1. Differential operations of vector analysis 253
1. Scalar and vector fields (253)
2. Vector fields and forms in R3 (253).
3. Differential operators grad, rot, div and V (256).
4. Some differential formulas of vector analysis (259).
5. Vector operations in curvilinear coordinates (261).

§ 2. Integral formulas of field theory 270
1. Classical integral formulas in vector notation (270).
2. Physical interpretation 273
3. Some further integral formulas (277)

§ 3. Potential fields
1. Potential of a vector field (281).
2. Necessary condition for potentiality (282).
3. Criterion for the potentiality of a vector field (288).
4. Topological structure of the region and potential (286).
5. Vector potential. Exact and closed forms (288).

§ 4. Application examples
1. Heat equation (295).
2. Equation of continuity (297).
3. Basic equations of the dynamics of a continuous medium (298).
4. Wave equation (300).

Chapter XV. Integration of differential forms on manifolds 305

§ 1. Some reminders from linear algebra
1. Algebra fdrm (305).
2. Algebra of skew-symmetric forms (306).
3. Linear mappings of linear spaces, and dual mappings of dual spaces (309). Tasks and exercises 310

§ 2. Variety.
1. Definition of a manifold (312).
2. Smooth manifolds and smooth mappings (317).
3. Orientation, manifolds and its boundaries (320).
4. Partition of the unity and realization of manifolds as surfaces in Rn (323).

§ 3. Differential forms and their integration on manifolds
1. Tangent space to a manifold at a point (329).
2. Differential form on a manifold (333).
3. External differential (335).
4. Integral of a form over a manifold (336).
5. Stokes formula (338).

§ 4. Closed and exact forms on a manifold
1. Poincaré's theorem (344).
2. Homology and cohomology 348

Chapter XVI. Uniform convergence and basic operations of analysis on series and families of functions 355

§ 1. Pointwise and uniform convergence
1. Pointwise convergence (355). 2. Statement of the main questions (356)
3. Convergence and uniform convergence of a family of functions depending on a parameter (358).
4. Cauchy criterion for uniform convergence (361).

§ 2. Uniform convergence of series of functions
1. Basic definitions and a criterion for the uniform convergence of the series (363).
2. The Weiergatrass criterion for the uniform convergence of the series (366).
3. Sign of Abel-Dirichlet (368).

§ 3. Functional properties of the limit function
1. Concretization of the problem (373).
2. Switching conditions for two passages to the limit (374).
3. Continuity and passage to the limit (376).
4. Integration and passage to the limit (380).
5. Differentiation and passage to the limit (381).

§ 4. Compact and dense subsets of the space of continuous functions
1. The Artsela-Ascoli theorem (391).
2. Metric space (393)
3. Stone's theorem (394).

Chapter XVII. Integrals depending on a parameter

§ 1. Eigenintegrals depending on a parameter
1. The concept of an integral depending on a parameter (400).
2. Continuity of an integral depending on a parameter (401).
3. Differentiation of an integral depending on a parameter (402).
4. Integration of an integral depending on a parameter (405)

§ 2. Improper integrals depending on a parameter
1. Uniform convergence of the improper integral with respect to the parameter (407).
2. Passing to the limit under the sign of an improper integral and the continuity of an improper integral depending on a parameter (415).
3. Differentiation of the improper integral with respect to the parameter (417).
4. Integration of the improper integral with respect to the parameter (420).

§ 3. Euler integrals
1. Beta function (428).
2. Gamma function 429
3. Relationship between functions C and D (432).
4. Some examples (433).

§ 4. Convolution of functions and initial information about generalized functions
1. Convolution in physical problems (leading considerations) (439).
2. Some general properties of convolution (442).
3. Delta-like families of functions and the Weierstrass approximation theorem (445).
4. Initial Views on distributions (450).

§ 5. Multiple integrals depending on a parameter
1. Own multiple integrals depending on the parameter (463).
2. Improper multiple integrals depending on a parameter (467).
3. Improper integrals with a variable singularity (469).
4. Convolution, fundamental solution, and generalized functions in the multidimensional case (473).

Chapter XVIII Reid Fourier and the Fourier Transform

§ 1. Basic general ideas associated with the concept of the Fourier series
1. Orthogonal systems of functions (488).
2. Fourier Coefficients 494
3. Fourier series 499
4. About one important source orthogonal systems of functions in analysis (506).

§ 2. Trigonometric Fourier Series
1. Basic types of convergence of the classical Fourier series (515)
2. Investigation of the pointwise convergence of the trigonometric Fourier series (520).
3. Smoothness of a function and the rate of decrease of the Fourier coefficients (530).
4. Completeness of the trigonometric system 535

§ 3. Fourier transform
1. Representation of a function by the Fourier integral (551).
2. Regularity of a function and the rate of decrease of its Fourier transform (562)
3. The most important hardware properties of the Fourier transform (566)
4. Application examples (572).

Chapter XIX. Asymptotic expansions

§ 1. Asymptotic formula and asymptotic series
1. Basic definitions (586).
2. General information on asymptotic series (591).
3. Power asymptotic series 696

§ 2. Asymptotic behavior of integrals (Laplace method)
1. The idea of ​​Laplace's method (602).
2. The principle of localization of the length of the Laplace integral (605).
3. Canonical integrals and their asymptotics 607
4. Main term of the asymptotics of the Laplace integral (610).
5. Asymptotic expansions of the Laplace integrals (613).

Brief summary of the book

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

 The text is provided with questions and tasks that complement the material of the book and existing problem books on analysis. An organic part of the text are examples of applications of the developed theory, which often serve as substantive problems of mechanics and physics.

 For university students studying in the specialty "Mathematics" and "Mechanics". It may be useful to students of faculties and universities with an extended program in mathematics, as well as specialists in the field of mathematics and its applications.

transcript

2 Mathematical analysis 1. Completeness: supremum and infimum of a numerical set. The principle of nested segments. The irrationality of the number The theorem on the existence of a limit of a monotone sequence. e number. 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 a segment. 5. Weierstrass' theorems on the largest and smallest 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. Investigation of a function using derivatives (monotonicity, extrema, convexity and inflection points, asymptotes). 10. Parametrically predefined functions and their differentiation. 11. Rolle, Lagrange and Cauchy theorems. 12. L'Hopital's rule. 13. Taylor's formula with a remainder term in the form of Lagrange. 14. Local Taylor formula with remainder term in Peano form. Expansion of basic elementary functions by the Taylor formula. 15. Riemann integrability criterion for a function. Classes of integrable functions. 16. The 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. Integration of rational fractions. 18. Methods of approximate calculation of definite integrals: methods of rectangles, trapezoids, parabolas. 19. Definite integral with variable upper limit; mean value theorems. 20. Geometric applications of a definite integral: the area of ​​a plane figure, the volume of a body in space. 21. Power series; expansion of functions in a 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 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. A necessary condition for a conditional extremum. Method of Lagrange multipliers. 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. The structure of the set of convergence 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. Reduction of a multiple integral to an iterated 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. A short course in mathematical analysis. V. 1, 2. Differential and integral calculus of functions of several variables. Harmonic analysis: a textbook for university students.- Ed. 3rd, revised - Moscow: Fizmatlit, p. 4. Mathematical analysis. T. 1.2: / ed. V.A. Course of mathematical analysis. T. 1, 2.- Ed. 4th, revised. and additional - Moscow: Nauka, p. 6. Ilyin, V.A. Fundamentals of mathematical analysis. Part 1, 2. - Ed. 4th, revised. and additional - Moscow: Nauka, p. Differential equations. 1. Existence and uniqueness theorem for the solution of the Cauchy problem for an ordinary differential equation of the first order. 2. Existence and uniqueness theorem for the solution of the Cauchy problem for an ordinary differential equation of the first order 3. Theorem on the continuous dependence of the solution of the Cauchy problem for an ordinary differential equation of the first order on parameters and initial data. 4. Differentiability theorem for the solution of the Cauchy problem for an ordinary differential equation of the first order with respect to parameters and initial data. 5. Linear ordinary differential equations (ODEs). General properties. Homogeneous ODE. Fundamental decision system. 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. Building 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 (ODEs). Fundamental decision system and fundamental matrix. Vronskian. Liouville formula. Structure common solution homogeneous system of ODEs. 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. Building 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 linear ordinary differential equation of the second order. 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. Existence and uniqueness theorem for a solution to a boundary value problem. 14. Autonomous systems. Solution properties. Singular points of a linear autonomous system of two equations. Stability and asymptotic stability in the sense of Lyapunov. Stability of a homogeneous system of linear differential equations with a variable matrix. 15. Stability in the First Approximation of a System of Nonlinear Differential Equations. Lyapunov's second method. References 1. Samoilenko, A.M. Differential equations: a practical course: a textbook for university students.- Ed. 3rd, revised. - Moscow: Higher School, p. 2. Agafonov, S.A. Differential equations: textbook. - 4th ed. 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. Philips, G. Differential equations: translation from English / G. Philips; edited by A.Ya. Khinchin. - 4th ed., ster. - Moscow: KomKniga, p. Algebra and number theory 1. Definition of a group, ring and field. Examples. Construction of the field of complex numbers. Raising to a power of complex numbers. Extracting the root from complex numbers. 2. Algebra of matrices. 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 (SLAE). SLAU research. Gauss method. Cramer's rule. 5. Ring of polynomials in one variable. Division theorem 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. Transition matrix from one basis to the second basis. 8. Subspaces. Operations on subspaces. Direct sum of subspaces. Criteria for the direct sum of subspaces. 9. Matrix rank. SLAU compatibility. The Kronecker-Capelli theorem. 10. Euclidean and unitary spaces. Metric concepts in Euclidean and unitary spaces. Cauchy-Bunyakovsky inequality. 11. Orthogonal systems of vectors. 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. Linear operator matrices in different bases.

5 14. Image and kernel, rank and defect of a linear operator. Dimension of the kernel and image. 15. Invariant subspaces of a linear operator. Eigenvectors and eigenvalues ​​of a linear operator. 16. A criterion for the diagonalizability of 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. Adjoint, normal operators and their simple properties. 19. Quadratic forms. Canonical and normal form of quadratic forms. 20. Constant sign quadratic forms, Sylvester's criterion. 21. The ratio of divisibility in the ring of integers. Division theorem with remainder. GCD and LCM of integers. 22. Continued (Continued) Fractions. Suitable fractions. 23. Prime numbers. Sieve of Eratosthenes. Infinity theorem prime numbers. Decomposition of a number into prime factors 24. Ant'e's function. multiplicative function. Möbius function. Euler function. 25. Comparisons. Basic properties. Complete billing system. The given system of deductions. Euler's and Fermat's theorems. 26. Comparisons of the first degree with one unknown. Comparison system of the first degree. Chinese remainder theorem. 27. Comparisons of any degree modulo composite. 28. Comparisons of the second degree. Legendre's symbol. 29. Primitive roots. 30. Indexes. Applying indices to solving comparisons. References 1. Kurosh, A.G. Lectures on general algebra: textbook / A.G. Kurosh. - 2nd ed., ster. - St. Petersburg: Publishing House "Lan", p. 2. Birkhoff, G. Modern applied algebra: textbook / Garrett Birkhoff, Thomas C. Barty; translation from English by Yu.I. Manina.- 2nd ed., erased- Saint 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, corrected - Moscow: FIZMATLIT, Vinogradov, I.M. Fundamentals of number theory: textbook.- Ed. 11th - St. Petersburg; Moscow; Krasnodar: Lan, p. 6. Bukhshtab, A.A. Number theory: textbook. - 3rd ed., stereotype. - St. Petersburg; Moscow; Krasnodar: Lan, p. 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. Mutual arrangement two straight lines. Angle between two lines. 3. Transformation of coordinates during the transition from one Cartesian coordinate system to another. 4. Polar, cylindrical and spherical coordinates. 5. Ellipse, hyperbola and parabola and their properties. 6. Classification of lines of the second order.

6 7. Equation of a plane defined in various ways. Mutual arrangement of two planes. The distance from a point to a plane. Angle between two planes. 8. Equations of a straight line in space. Mutual arrangement of two straight lines, a straight line and a plane. The distance from a point to a line. The angle between two lines, a line and a plane. 9. Ellipsoids, hyperboloids and paraboloids. Rectilinear generators of surfaces of the second order. 10. Surfaces of revolution. Cylindrical and conical surfaces. 11. Definition of an elementary curve. Ways to set a curve. Curve length (definition and calculation). 12. Curvature and torsion of a curve. 13. Accompanying frame of a smooth curve. Frenet formulas. 14. The first quadratic form of a smooth surface and its applications. 15. The second quadratic form of a smooth surface, the normal curvature of the surface. 16. Principal directions and principal surface curvatures. 17. Lines of curvature and asymptotic lines of a surface. 18. Average and Gaussian curvature of a surface. 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. At 2 o'clock, a study guide. - 2nd ed. - Novosibirsk: Siberian University Publishing House, p. 4. Efimov, N.V. A short course in analytic 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 for 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. The content of teaching mathematics in high school. 2. Didactic principles of teaching mathematics. 3. Methods scientific knowledge. 4. Visibility in teaching mathematics. 5. Forms, methods and means of monitoring and evaluating the knowledge and skills of students. Marking standards. 6. Extracurricular work mathematics. 7. Mathematical concepts and methods of their formation. 8. Tasks as a means of teaching mathematics. 9. In-depth study of mathematics: content, methods and forms of organization of education. 10. Types of mathematical judgments: axiom, postulate, theorem.

7 11. Summary of the lesson in mathematics. 12. Lesson of mathematics. Types of lessons. Lesson analysis. 13. The study of mathematics in a small school: content, methods and forms of organization of education. 14. New learning technologies. 15. Differentiation of teaching mathematics. 16. Individualization of teaching mathematics. 17. Motivation learning activities schoolchildren. 18. Logical and didactic analysis of the topic. 19. Technological approach to teaching mathematics 20. Humanization and humanitarization of teaching mathematics. 21. Education in the process of teaching mathematics. 22. Methods of studying identical transformations. 23. Methods for studying inequalities. 24. Methods of studying the function. 25. Methods for studying the topic "Equations and inequalities with a module." 26. Methods for studying the topic "Cartesian coordinates". 27. Methods of studying polyhedra and round bodies. 28. Methods for studying the topic "Vectors". 29. Methods for solving problems for movement. 30. Methods for solving problems for joint work. 31. Methodology for studying the topic "Triangles" 32. Methodology for studying the topic "Circle and circle". 33. Methods for solving problems for alloys and mixtures. 34. Methods for studying the topic "Derivative and Integral". 35. Methodology for studying the topic "Irrational equations and inequalities." 36. Methods for studying the topic "Solving equations and inequalities with parameters." 37. Methods of studying the basic concepts of trigonometry. 38. Methods for studying the topic "Trigonometric equations" 39. Methods for studying the topic "Trigonometric inequalities". 40. Methodology for studying the topic “Reverse trigonometric functions". 41. Methodology for studying the topic " General Methods solving equations in the school course of mathematics. 42. Methodology for studying the topic " Quadratic equations". 43. Methods for studying the basic concepts of stereometry 44. Methods for studying the topic "Ordinary fractions." 45. Methods for studying the topic "Use of the derivative in the study of functions" Literature 1. Argunov, B.I. School course mathematics and methods of teaching it. - Moscow: Education, p. 2. Zemlyakov, A.N. Geometry in the 11th grade: methodological recommendations for studies. A.V. Pogorelova: a guide for a teacher. - 3rd ed., Dor. - M .: Education, p. 3. The study of algebra in grades 7-9: a book for the teacher / Yu.M. Kolyagin, Yu.V. Sidorov, M.V. Tkacheva and others - 2nd ed. 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) of physical and mathematical education. - Moscow: Drofa, 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. A necessary condition for a conditional extremum. Method of Lagrange multipliers. 27. Number series. Cauchy Convergence Criterion

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

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

MINISTRY OF EDUCATION AND SCIENCE OF RUSSIA Federal State Budgetary Educational Institution of Higher Education "Chelyabinsk State University" admission committee,

EAST KAZAKHSTAN STATE TECHNICAL UNIVERSITY IM. D. SERIKBAYEVA Faculty of Information Technology and Business APPROVED by the Dean of FITIB N.Denisova 2016 ENTRANCE EXAMS PROGRAM

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

Ministry of Education and Science of the Russian Federation Ivanovo State University Faculty of Mathematics and computer science PROGRAM OF ENTRANCE EXAMINATION TO MASTER'S STUDIES FOR EDUCATION IN

EAST KAZAKHSTAN STATE TECHNICAL UNIVERSITY IM. D. SERIKBAYEVA Faculty of Information Technology and Business APPROVED by Dean of FITIB N.Denisova 2016 ENTRANCE EXAMS PROGRAM

Annotation to the work program of the discipline Author Fedorov Yu.I., Associate Professor Name of the discipline: B1.B.05Mathematics

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 examination questions: 1 semester 1. Sets and operations on them. 2. Cartesian product of sets. 3. Limit points. 4. Sequence limit. 5. Function limit. 6. Infinitely small.

“APPROVED” Acting Director of the FMITI Pop E.N. MATHEMATICS, master's program "Complex Analysis"

Methodical materials for teachers. Exemplary plans for lectures. 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. Actions on matrices 2. Determinants 2.1. Basic concepts 2.2. Properties of determinants 3. Nondegenerate 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. Nondegenerate matrices 3.1.

I APPROVE Department of Physical and Mathematical Disciplines E.N. Kiryukhova 20 g, protocol Questions for the exam in the discipline "Mathematics" Specialties " Information Systems and technologies" absentee form receiving

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

4. Annotation to the work program of the discipline Author Fedorov Yu.I., Associate Professor Name of the discipline: B1.B.04 Higher mathematics

1. The purpose and objectives of the discipline Mathematical Analysis

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

NAN CHOU VO Academy of Marketing and Social Information Technologies ABSTRACT OF THE EDUCATIONAL DISCIPLINE Direction of preparation 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 IM. D. SERIKBAYEV Faculty of Information Technology and Energy APPROVED 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 degree in the specialty "6M070500-Mathematical and computer modeling" Mathematical analysis I, II, III 1. Completeness: the existence of a limit of a monotone sequence.

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

FEDERAL AGENCY FOR EDUCATION State Educational Institution of Higher Professional Education “Ural State University. A.M. Gorky "Mathematics - Mechanical

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

Ministry of Education and Science of the Krasnodar Territory State Budgetary Vocational Educational Institution of the Krasnodar Territory "Krasnodar Information Technology College" lesson

Ministry of Education and Science of the Russian Federation Yaroslavl State Pedagogical University them. K.D. Ushinsky» U T V E R ZH D A Yu First Vice-Rector M.V. Novikov 20 PROGRAM

The 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" mathematical disciplines

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

Appendix 3 MINISTRY OF SCIENCE AND EDUCATION OF THE RUSSIAN FEDERATION 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 training sessions in the discipline of mathematical analysis Index of specialty NF course I semester 1 Leading discipline Candidate of Physical and Mathematical Sciences, Associate Professor Budochkina

Annotation to the work program of the discipline B1.B.4 Mathematics Direction of training Profile of training 05.03.01 Geology Geophysics Qualification (degree) of the graduate 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 MP Vishnevsky Lecturer: 1st semester 1. Introduction. Sets and operations on them. Set mappings. Countable sets. Valid

The program of the entrance exam to the master's program in the specialty "6M060100-MATHEMATICS" Mathematical analysis Numerical function and methods for its assignment. Limit of a function and main theorems, definitions. Criteria

ENTRY TEST PROGRAM educational program higher education program for the training of scientific and pedagogical personnel in the postgraduate course of the FSBEI HE "Oryol State University named after

QUESTIONS AND TYPICAL TASKS for the final exam in the discipline "Mathematical Analysis" Applied Mathematics At the oral exam, the student receives two theoretical questions and two tasks Total 66 questions per year

annotation work program disciplines Mathematical analysis (name of discipline) Direction of training 03.03.02 physics Training profile "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

Program of the course "Mathematical Analysis". Semester 1 (72 hours of lectures, 72 hours of practical training) Thematic plan lectures. I. Introduction to analysis. 1. Elements of set theory. 2. Integers. 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 tasks .. What is a numerical

Matrices. Algebra and geometry 1. Determinants. Decomposition of the determinant in row and column. Algebra 2. Geometric vectors. Scalar product of vectors. Vector and mixed product of vectors.

Approved at the meeting of the department "Mathematics and Informatics" Protocol 2 (25) "8" September 2015. head Department of Ph.D. Timshina D.V. Questions for the test in the discipline "LINEAR ALGEBRA AND MATHEMATICAL ANALYSIS"

Funds Funds of evaluation funds for discipline B.2.1 "Mathematical analysis" for current control progress and intermediate certification of students in the direction 080100.62 "Economics"

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

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

1 2 1. GOALS AND OBJECTIVES OF PRACTICAL LESSONS Practical classes in the discipline "Mathematics" are held with the aim of: 1. Formation of skills: - to systematize the knowledge and practical

State Committee RSFSR for science and high school SIBERIAN STATE GEODETIC ACADEMY V.P. Verbnaya D.A. KRYMSKIH E.S. PLYUSNINA HIGHER MATHEMATICS Toolkit for students

GBOU SPO Prokopyevsk Polytechnic College PROGRAM OF EDUCATIONAL DISCIPLINE "ELEMENTS OF HIGHER MATHEMATICS" Recommended for specialty 30111 Computer networks Name of qualification basic training

SHORT PROGRAM OF ENTRANCE EXAMS TO THE MASTER'S STATE PROGRAM "MATHEMATICAL EDUCATION" 2015 Section 1. Algebra and number theory 1. Algebraic and trigonometric forms of a complex number.

The program of the written exam in "Higher Mathematics" for the 1st year correspondence departments Faculty of Economics during the winter session, the written exam is held for two hours. In the exam for each student

EVALUATION TOOLS FOR CURRENT CONTROL OF PROGRESS, INTERIM CERTIFICATION ON THE RESULTS OF MASTERING THE DISCIPLINE Academic discipline B.2.1 - Mathematics Training profile: Production management Topics

federal agency in education SEI HPE "Pomor State University named after M. V. Lomonosov" APPROVED by the Rector of Pomor state university named after M.V. Lomonosova I.R. Lugovskaya

QUESTIONS FOR PREPARATION FOR THE EXAM Vector algebra and analytic geometry. Vector definition. Vector equality. Linear operations on vectors. Linear dependence of vectors. Basis and coordinates.

2 Intermediate attestation 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

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 physical and mathematical specialties. It may be useful to 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 of the course of classical analysis with modern mathematical courses (algebra, differential geometry, differential equations, complex and functional analysis).

The 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. Remarks on the proofs.
    • 3. Some special designations.
    • 4. Concluding remarks.
  • § 2. Sets and elementary operations on sets
    • 1. The concept of a 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 mutually inverse mappings.
    • 4. Function as relation. Function graph.
  • § 4. Some additions
    • 1. Power 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 (real) numbers
  • § 1. Axiomatics and some general properties of a set real numbers
    • 1. Definition of the set of real numbers.
    • 2. Some general algebraic properties of real numbers.
    • 3. The axiom of completeness and the existence of an upper (lower) bound of a number 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. Principle of Archimedes.
    • 4. Geometric interpretation of the set of real numbers and computational aspects of operations with real numbers.
  • § 3. Main lemmas related to the completeness of the set of real numbers
    • 1. Lemma on nested segments (Cauchy-Cantor principle).
    • 2. Finite cover lemma (Borel-Lebesgue principle.
    • 3. Lemma about limit point(principle of Bolzano-Weierstrass).
  • § 4. Countable and uncountable sets
    • 1. Countable sets.
    • 2. Power of the continuum.
• Chapter III. Limit
  • § 1. Limit of a sequence
    • 1. Definitions and examples.
    • 2. Properties of the sequence limit.
    • 3. Existence questions for the limit of a sequence.
    • 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 (base limit).
    • 4. Questions of 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. Points of break.
  • § 2. Properties of continuous functions
    • 1. Local properties.
    • 2. Global properties of continuous functions.
• Chapter V. Calculus of Differentials
  • § 1. Differentiable function
    • 2. A function differentiable at a point.
    • 3. Tangent; geometric sense 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 the main elementary functions.
    • 5. Differentiation of the simplest implicitly given function.
    • 6. Derivatives of higher orders.
  • § 3. Basic theorems of differential calculus
    • 1. Fermat's Lemma and Rolle's Theorem.
    • 2. Lagrange and Cauchy theorems on finite increment.
    • 3. Taylor formula.
  • § 4. Investigation of functions by methods of differential calculus
    • 1. Conditions for the monotonicity of a function.
    • 2. Conditions for the internal extremum of the function.
    • 3. Conditions for the convexity of a function.
    • 4. L'Hopital's rule.
    • 5. Plotting a function graph.
  • § 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. The fall of bodies in the atmosphere.
    • 5. Once again about the number e and the function.
    • 6. Fluctuations.
  • § 7. Antiderivative
    • 1. Antiderivative and indefinite integral.
    • 2. Basic general tricks finding the original.
    • 3. Antiderivatives of rational functions.
    • 4. Primitives of the species.
    • 5. Primitives of the species.
• Chapter VI. Integral
  • § 1. Definition of the integral and description of the set of integrable functions
    • 1. Problem and leading considerations.
    • 2. Definition of the Riemann integral.
    • 3. Set of integrable 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 interval 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 the Taylor formula.
    • 4. Change of variable in the integral.
    • 5. Some examples.
  • § 4. Some applications of the integral
    • 1. Additive function of an oriented interval and an integral.
    • 2. Path length.
    • 3. Area of ​​a curvilinear trapezoid.
    • 4. Volume of the body of revolution.
    • 5. Work and energy.
  • § 5. Improper integral
    • 1. Definitions, examples and basic properties of improper integrals.
    • 2. Investigation 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. Compact spaces in R m .
    • Tasks and exercises.
  • § 2. Limit and continuity of a function of several variables
    • 1. Function limit.
    • 2. Continuity of a function of several variables and properties of continuous functions.
• Chapter VIII. Differential calculus of functions of several variables
  • § 1. Linear structure in Rm
    • 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. Jacobi matrix.
    • 4. Continuity, partial derivatives, and differentiability of a function at a point.
  • § 3. Basic laws of differentiation
    • 1. Linearity of the operation of differentiation.
    • 2. Differentiation of the composition of mappings.
    • 3. Differentiation of the inverse mapping.
  • § 4. Basic facts of the differential calculus of real-valued functions of several variables
    • 1. The mean value theorem.
    • 2. A sufficient condition for the differentiability of a function of several variables.
    • 3. Partial derivatives of higher order.
    • 4. Taylor formula.
    • 5. Extrema of functions of several variables.
    • 6. Some geometric images associated with functions of several variables.
  • § 5. The implicit function theorem
    • 1. Statement of the question and leading considerations.
    • 2. The simplest version of the implicit function theorem.
    • 3. Transition to the case of dependence F(x 1 , …, x n , y) = 0.
    • 4. The implicit function theorem.
  • § 6. Some consequences of the implicit function theorem
    • 1. Inverse function theorem.
    • 2. Local reduction of a smooth mapping to canonical form.
    • 3. Dependence of functions.
    • 4. Local decomposition of a diffeomorphism into a composition of the simplest.
    • 5. Morse's lemma.
  • § 7. Surface in R n and the theory of conditional extremum
    • 1. A surface of dimension k in R n .
    • 2. Tangent space.
    • 3. Conditional extremum.
• Some tasks of colloquia
• 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 a metric space.
    • 4. Direct product of metric spaces.
  • § 2. Topological space
    • 1. Basic definitions.
    • 2. Subspace of a topological space.
    • 3. Direct product of topological spaces.
  • § 3. Compacta
    • 1. Definition and general properties of a compactum.
    • 2. Metric compacts.
  • § 4. Connected topological spaces
  • § 5. Complete metric spaces
    • 1. Basic definitions and examples.
    • 2. Completion of a metric space.
  • § 6. Continuous mappings of topological spaces
    • 1. Display limit.
    • 2. Continuous mappings.
  • § 7. The principle of contraction mappings
• Chapter X. 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. Norm of the operator.
    • 3. The space of continuous operators.
  • § 3. Mapping differential
    • 1. A map that is differentiable at a point.
    • 2. General laws of differentiation.
    • 3. Some examples.
    • 4. Partial derivatives of mappings.
  • § 4. Finite increment theorem and some examples of its use
    • 1. Finite increment theorem.
    • 2. Some examples of application of the finite increment theorem.
  • § 5. Derivative mappings of higher orders
    • 1. Definition of the nth differential.
    • 2. Derivative with respect to the vector and calculation of the values ​​of the nth differential.
    • 3. Symmetry of differentials of higher order.
    • 4. Some remarks.
  • § 6. The Taylor formula and the study of extrema
    • 1. Taylor formula for mappings.
    • 2. Study of internal extremes.
    • 3. Some examples.
  • § 7. General implicit function theorem
• Chapter XI. Multiple integrals
  • § 1. The Riemann integral on an n-dimensional interval
    • 1. Definition of the integral.
    • 2. Lebesgue criterion for the integrability of a function in the sense of Rnman.
    • 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 an iterated 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 - change of variables.
    • 2. Measurable sets and smooth mappings.
    • 3. One-dimensional case.
    • 4. The case of the simplest diffeomorphism in R n .
    • 5. Composition of mappings and the change of variables formula.
    • 6. Additivity of the integral and completion of the proof of the formula for the change of variables in the integral.
    • 7. Some consequences and generalizations of the formula for the change of variables in multiple integrals.
  • § 6. Improper multiple integrals
    • 1. Basic definitions.
    • 2. Majorant Approach to the Convergence of an Improper Integral.
    • 3. Change of variables in the improper integral.
• Chapter XII. Surfaces and differential forms in R n
  • § 1. Surfaces in R n
  • § 2. Surface orientation
  • § 3. Surface edge and its orientation
    • 1. Surface with edge.
    • 2. Coordination of surface and edge orientation.
  • § 4. Surface area in Euclidean space
  • § 5. Introduction to differential forms
    • 1. Differential form, definition and examples.
    • 2. Coordinate notation of the differential form.
    • 3. External shape differential.
    • 4. Transfer of vectors and shapes in mappings.
    • 5. Forms on surfaces.
• Chapter XIII. Curvilinear and surface integrals
  • § 1. Integral of differential form
    • 1. Initial tasks, suggestive considerations, examples.
    • 2. Definition of the integral of the shape over an oriented surface.
  • § 2. Volume form, integrals of the first and second kind
    • 1. The mass of the material surface.
    • 2. Area of ​​the surface as an integral of the shape.
    • 3. The shape of the volume.
    • 4. Expression of the shape of the volume 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 R 3 .
    • 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 R 3 .
    • 3. Differential operators grad, rot, div and V.
    • 4. Some differential formulas of vector analysis.
    • 5. Vector operations in curvilinear coordinates.
  • § 2. Field theory integral formulas
    • 1. Classical integral formulas in vector notation.
    • 2. Physical interpretation.
    • 3. Some further integral formulas.
  • § 3. Potential fields
    • 1. Potential of a vector field.
    • 2. A necessary condition for potentiality.
    • 3. Criterion for the potentiality of a vector field.
    • 4. Topological structure of the region and potential.
    • 5. Vector potential. Exact and closed forms.
  • § 4. Application examples
    • 1. Equation of heat conduction.
    • 2. Equation of continuity.
    • 3. Basic equations of the dynamics of a continuous medium.
    • 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 dual mappings of dual spaces. Tasks and exercises
  • § 2. Variety.
    • 1. Definition of variety.
    • 2. Smooth manifolds and smooth mappings.
    • 3. Orientation, manifolds and its boundaries.
    • 4. Partitioning of the Unity and Realization of Manifolds as Surfaces in R n .
  • § 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 a form over a manifold.
    • 5. Stokes formula.
  • § 4. Closed and exact forms on a manifold
    • 1. Poincaré's theorem.
    • 2. Homology and cohomology.
• Chapter XVI. Uniform convergence and basic operations of analysis on series and families of functions
  • § 1. Pointwise and uniform convergence
    • 1. Pointwise convergence.
    • 2. Statement of the main 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 a criterion for the uniform convergence of a series.
    • 2. The Weiergatrass criterion for the uniform convergence of the series.
    • 3. Sign of Abel-Dirichlet.
  • § 3. Functional properties of the limit function
    • 1. Specification of the task.
    • 2. Conditions for commutation of two passages to the limit.
    • 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 Artsela-Ascoli theorem.
    • 2. Metric space.
    • 3. Stone's theorem.
• Chapter XVII. Integrals depending on a parameter
  • § 1. Eigenintegrals depending on a parameter
    • 1. The concept of an integral depending on a parameter.
    • 2. Continuity of an integral depending on a 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. Passing to the limit under the sign of an improper integral and the continuity of an improper integral depending on a parameter.
    • 3. Differentiation of the improper integral with respect to a parameter.
    • 4. Integration of the improper integral with respect to a parameter.
  • § 3. Euler integrals
    • 1. Beta function.
    • 2. Gamma function.
    • 3. Relationship between functions C and D.
    • 4. Some examples.
  • § 4. Convolution of functions and initial information about generalized functions
    • 1. Convolution in physical problems (leading considerations).
    • 2. Some general properties of convolution.
    • 3. Delta-like families of functions and the Weierstrass approximation theorem.
    • 4. Initial ideas about distributions.
  • § 5. Multiple integrals depending on a parameter
    • 1. Own multiple integrals depending on the parameter.
    • 2. Improper multiple integrals depending on a parameter.
    • 3. Improper integrals with a variable singularity.
    • 4. Convolution, fundamental solution and generalized functions in the multidimensional case.
• Chapter XVIII Reid Fourier and the Fourier Transform
  • § 1. Basic general ideas related to the concept of a Fourier series
    • 1. Orthogonal systems of functions.
    • 2. Fourier coefficients and Fourier series.
    • 3. On 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. Investigation of the pointwise convergence of the trigonometric Fourier series.
    • 3. Smoothness of the function and 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 a function and 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 method)
    • 1. The idea of ​​the Laplace method.
    • 2. The principle of localization of the length of the Laplace integral.
    • 3. Canonical integrals and their asymptotics.
    • 4. Leading term of the asymptotics of the Laplace integral.
    • 5. Asymptotic expansions of the Laplace integrals.
• Tasks and exercises
• Literature
• Index of main symbols
• Alphabetical index

The textbook is the first part of a three-volume course on mathematical analysis for higher educational institutions of the 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, the differential and integral calculus of functions of one variable and their applications, the differential calculus of functions of many variables, and the theory of implicit functions.

REAL NUMBERS.
In the previous chapter, we saw that the development of the theory of real numbers is necessary for a rigorous and consistent study of the concept of a limit, which is one of the most important concepts mathematical analysis.

The theory of real numbers we need, which is presented in this chapter, includes the definition of the operations of ordering the addition and multiplication of these numbers and the establishment of the basic properties of these operations, as well as the proof of the existence of exact edges for sets of numbers bounded above or below.

At the end of the chapter, an idea is given of additional questions in the theory of real numbers that are not necessary for the construction of the theory of limits and, in general, the course of mathematical analysis (the completeness of the set of real numbers in the sense of Hilbert, the axiomatic construction of the theory of real numbers, the connection between various methods of introducing real numbers).

Free download e-book in a convenient format, watch and read:
Download the book Mathematical analysis, Initial course, Ilyin V.A., Sadovnichiy V.A., Sendov B.X., Tikhonov A.N., 1985 - fileskachat.com, fast and free download.

• Mathematical analysis, Continuation of the course, Ilyin V.A., Sadovnichiy V.A., Sendov B.Kh., Tikhonov A.N., 1987
• Mathematical analysis, Primary course, Part 1, Ilyin V.A., Sadovnichiy V.A., Sendov Bl.Kh., 1985
• Mathematical Analysis, Primary Course, Volume 1, Ilyin V.A., Sadovnichiy V.A., Sendov B.Kh., 1985
• Mathematical analysis - Ilyin V.A., Sadovnichiy V.A., Sendov Bl.Kh. - Continuation of the course

The following tutorials and books.