Mathematical analysis, functional analysis. Mathematical analysis Mathematical analysis pdf

T. 1. Differential and integral calculus of functions of one variable.

T. 2. Rows. Differential and integral calculus of functions of several variables.

V. 3. Harmonic analysis. Elements of functional analysis.

Moscow: Bustard; v.1- 2003, 704 pages; v.2- 2004, 720 pages; v.3- 2006, 351s.

The textbook corresponds to the new program for universities. Particular attention in the textbook is paid to the presentation of qualitative and analytical methods; it also reflects some geometric applications of analysis. It is intended for students of universities and physical and mathematical, and engineering and physical specialties of technical universities, as well as students of other specialties for in-depth mathematical training.

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Volume 1. Table of contents
Preface 3
Introduction 7
Chapter 1
Differential calculus of functions of one variable
§ 1. Sets and functions. Logic Symbols 13
1.1. Sets. Operations on sets 13
1.2*. Functions 16
1.3*. Finite sets and natural numbers.
1.4. Groupings of elements of a finite set 29
1.5. Logic Symbols 33
§ 2. Real numbers 35
2.1. Properties of Real Numbers 35
2.2*. Properties of addition and multiplication 39
2.3*. Order properties 47
2.4*. Continuity property of real numbers 51
2.5*. Sections in the set of real numbers 52
2.6*. Rational powers of real numbers 58
2.7. Newton's binomial formula 60

§ 3. Numerical sets 63
3.1. Extended number line 63
3.2. Intervals of real numbers. Neighborhood 64
3.3. Bounded and unbounded sets 68
3.4. Upper and lower bounds of number sets 70
3.5*. Arithmetic Properties top and bottom faces... 75
3.6. Archimedes principle 78
3.7. The principle of nested segments 80
3.8*. Uniqueness of a continuous ordered field.... 85
§ 4. Limit of a numerical sequence 92
4.1. Determining the Limit of a Number Sequence 92
4.2. Uniqueness of the limit of a numerical sequence... 100
4.3. Passing to the limit in inequalities 101
4.4. Boundedness of convergent sequences 107
4.5. Monotone sequences 108
4.6. Bolzano-Weierstrass theorem 113
4.7. Cauchy criterion for sequence convergence 115
4.8. Infinitesimal Sequences 118
4.9. Limit properties associated with arithmetic operations on sequences 120
4.10. Depiction of real numbers by infinite decimals 133
4.11*. Countable and uncountable sets 141
4.12*. Upper and Lower Sequence Limits 149
§ 5. Limit and continuity of functions 153
5.1. Valid Functions 153
5.2. Function Setting Methods 156
5.3. Elementary functions and their classification 160
5.4. First definition of function limit 162
5.5. Continuous functions 172
5.6. The condition for the existence of a function limit 177
5.7. Second definition of function limit 179
5.8. The limit of the set union function 184
5.9. One-sided limits and one-sided continuity... 185
5.10. Function limits properties 189
5.11. Infinitely Small and Infinitely Large Functions 194
5.12. Various forms of continuity notation
5.13. Classification of breakpoints of a function 202
5.14. Limits of Monotonic Functions 204
5.15. Cauchy criterion for the existence of a limit of a function 210
5.16. Limit and continuity of the composition of functions 212
§ 6. Properties of continuous functions on intervals 216
6.1. Boundedness of continuous functions. Reachability of extreme values 216
6.2. Intermediate values of continuous functions 218
6.3. Inverse functions 221
6.4. Uniform continuity. Continuity modulus.... 228
§ 7. Continuity of elementary functions 235
7.1. Polynomials and Rational Functions 235
7.2. Exponential, logarithmic and power functions. . 236
7.3. Trigonometric and inverse trigonometric functions 246
7.4. Continuity of elementary functions 248
§ 8. Comparison of functions. Calculating Limits 248
8.1. Some Remarkable Limits 248
8.2. Function comparison 253
8.3. Equivalent Functions 264
8.4. The method of extracting the main part of a function and its application to the calculation of limits 267
§ 9. Derivative and differential 271
9.1. Definition of a derivative 271
9.2. Function differential 274
9.3. The geometric meaning of the derivative and differential ... 280
9.4. The physical meaning of the derivative and differential 284
9.5. Rules for calculating derivatives related to arithmetic operations on functions 288
9.6. Derivative of the inverse function 291
9.7. Derivative and Differential of a Complex Function 294
9.8. Hyperbolic functions and their derivatives 301
§ten. Derivatives and differentials of higher orders 304
10.1. Higher order derivatives 304
10.2. Higher order derivatives sums and products of functions 306
10.3. Higher-order derivatives of complex functions, of inverse functions, and of functions given
10.4. Higher order differentials 311
§eleven. Mean value theorems for differentiable functions 313
11.1 Fermat's Theorem

11.2. Rolle, Lagrange and Cauchy's mean value theorems. . 316
§12. Disclosure of uncertainties according to L'Hopital's rule 327
12.1 Uncertainties of the form 0/0
12.2 Uncertainties of the form ----

12.3. Generalization of L'Hopital's Rule 337
§ 13. Taylor formula 339
13.1. Derivation of the Taylor Formula 339
13.2. Taylor polynomial as a polynomial of the best approximation of a function in a neighborhood of a given point 344
13.3. Taylor formulas for basic elementary
13.4. Calculating Limits Using the Taylor Formula (Principal Part Extraction Method) 351
§ 14. Investigation of the behavior of functions 353
14.1. Function Monotonicity Test 353
14.2. Finding the largest and the smallest values functions 356
14.3. Bulge and inflection points 365
14.5. Plotting Functions 377
§ 15. Vector function 387
15.1. The concept of limit and continuity for a vector function 387
15.2. Derivative and Differential of a Vector Function 391
§ 16. Curve length 397
16.3. Curve orientation. Arc curve. Sum of curves. Implicit Curves 408
16.4. Tangent to a curve. The geometric meaning of the derivative of a vector function 411
16.7. The physical meaning of the derivative of a vector function... 425
§17. Curvature and Torsion of a Curve 426
17.1. Two lemmas. Radial and transverse velocity components 426
17.2. Determining the curvature of a curve and calculating it 430
17.3. Main normal. Contact plane 434
17.4. Center of Curvature and Evolute of a Curve 436
17.5. Formulas for curvature and evolute of a plane curve.... 437
17.6. Involute 444
17.7. Torsion of a spatial curve 447
17.9. Formulas for calculating torsion 451
Chapter 2
Integral calculus of functions of one variable
§eighteen. Definitions and properties of the indefinite integral 453
18.1. Antiderivative and Indefinite Integral 453
18.2. Basic properties of the integral 456
18.3. Table integrals 458
18.4. Substitution Integration (Change of Variable) 461
18.5. Integration by parts 464
18.6*. Generalization of the concept of antiderivative 467
§ 19. Some information about complex numbers and polynomials. . 473
19.1. Complex numbers 473
19.2*. Formal theory of complex numbers 481
19.3. Some concepts of analysis in the field of complex numbers 482
19.4. Factoring polynomials 486
19.5*. Greatest Common Divisor of Polynomials 490
19.6. Decomposition of Proper Rational Fractions into Elementary Fractions 495
§ 20. Integration of rational fractions 503
20.1. Integration of elementary rational fractions... 503
20.2. General case 506
20.3*. Ostrogradsky method 508
§21. Integration of some irrationalities 514
21.1. Preliminary remarks 514
21.2. Integrals of the form \R\X, [^jf , ... , (^if]<** 515
21.3. Integrals of the form \Wx, Jax2 + bx + c) dx. Euler substitutions 518
21.4. Integrals of differential binomials 522
21.5. Integrals of the form n" " Jax2 + bx + c
§ 22. Integration of some transcendental functions.... 526
22.1. Type integrals JR(sin x,cosx)dx 526
22.2. Integrals of the form Jsinm x cos" x dx 528
22.3. Integrals of the form Jsin ax cos |3x dx 530
22.4. Integrals of transcendental functions calculated by integration by parts. . 530
22.5. Integrals of the form J.R(sh x, ch x) dx 532
22.6. Remarks on integrals not expressible in terms of elementary functions 532
§ 23. Definite integral 533
23.1. Definition of the Riemann integral 533
23.2*. Cauchy criterion for the existence of an integral 539
23.3. Boundedness of the integrable function 541
23.4. Upper and lower Darboux sums. Upper and lower Darboux integrals 543
23.5. Necessary and sufficient conditions for integrability. . 547
23.6. Integrability of continuous and monotone functions. 548
23.7*. Integrability criteria for Darboux and Riemann 551
23.8*. Function fluctuations 556
23.9*. Dubois-Reymond integrability criterion 563
23.10*. Lebesgue integrability criterion 566
§ 24. Properties of integrable functions 570
24.1. Properties of the Definite Integral 570
24.2. First mean value theorem for a definite integral 583
§25. Definite Integral with Variable Limits
25.1. Continuity of the integral over the upper limit
25.2. Differentiability of the integral with respect to the upper limit of integration. The existence of an antiderivative of a continuous function 588
25.3. Newton-Leibniz formula 591
25.4*. Existence of a generalized antiderivative. The Newton-Leibniz formula for the generalized antiderivative. . 592
§26. Formulas for Change of Variable in an Integral and Integration by Parts 596
26.1. Variable substitution 596
26.2. Integration by parts 600
26.3*. The second mean value theorem for a certain
26.4. Integrals of vector functions 606
§27. Measure of flat open sets 608
27.1. Determining the measure (area) of an open set 608
27.2. Measure properties of open sets 612
§28. Some Geometrical and Physical Applications of the Definite Integral 618
28.1. Area Calculation 618
28.2*. Hölder and Minkowski integral inequalities... 625
28.3. The volume of the body of rotation 630
28.4. Curve Length Calculation 632
28.5. Surface area of rotation 637
28.6. Work force 640
28.7. Calculation of static moments and coordinates of the center of gravity of a curve 641
§ 29. Improper integrals 644
29.1. Definition of Improper Integrals 644
29.2. Integral Calculus Formulas for Improper Integrals 652
29.3. Improper Integrals of Nonnegative Functions 657
29.4. Cauchy criterion for the convergence of improper integrals. 665
29.5. Absolutely convergent integrals 666
29.6. Investigation of the convergence of integrals 671
29.7. Asymptotic Behavior of Integrals with Variable Limits of Integration 677
Index 685
Index of basic symbols 695

Volume 2. Table of contents
Preface 3
Chapter 3

ranks
§ 30. Number series 5
30.1. Series definition and convergence 5
30.2. Properties of convergent series 9
30.3. Cauchy criterion for series 11 convergence
30.4. Series with non-negative members 13
30.5. Comparison test for series with non-negative members. Method for extracting the main part of a member of series 16
30.6. d'Alembert and Cauchy tests for series with non-negative terms 20
30.7. Integral criterion for the convergence of series with non-negative terms 23
30.8*. Hölder and Minkowski inequalities for finite and infinite sums 25
30.9. Alternating Series 27
30.10. Absolutely convergent series. Application of absolutely convergent series to the study of convergence
30.11. d'Alembert's and Cauchy's signs for arbitrary number series 38
30.12. Convergent series that do not converge absolutely. Riemann's Theorem 39
30.13. Abel transformation. Convergence criteria for Dirichlet and Abel 43
30.14*. Asymptotic behavior of residuals of convergent series and partial sums of divergent series 48
30.15. On the summability of series by the method of arithmetic means 52
§ 31. Infinite products 53
31.1. Basic definitions. The simplest properties of infinite products 53
31.2. Cauchy criterion for the convergence of infinite products 57
31.3. Infinite products with real
31.4. Absolutely convergent infinite products... 62
31.5*. Riemann Zeta Function and Prime Numbers 65
§ 32. Function sequences and series 67
32.1. Convergence of functional sequences
32.2. Uniform convergence of functional sequences 71
32.3. Uniformly Converging Function Series 79
32.4. Properties of uniformly convergent series and sequences 90
§ 33. Power series 100
33.1. Radius of convergence and circle of convergence of the power series 100
33.2*. Cauchy-Hadamard formula for convergence radius
33.3. Analytic functions 110
33.4. Analytic Functions in the Real Domain... 112
33.5. Expansion of functions into power series. Different ways to write the remainder term of the Taylor formula. . 116
33.6. Expansion of elementary functions in a Taylor series... 121
33.7. Methods for expanding functions into power series 131
33.8. Sterling Formula 138
33.9*. Formula and Taylor series for vector functions 141
33.10*. Asymptotic power series 143
33.11*. Properties of asymptotic power series 149
§ 34. Multiple series 153
34.1. Multiple number series 153
34.2. Multiple function series 162
Chapter 4
Differential calculus of functions of several variables
§ 35. Multidimensional spaces 165
35.1. Neighborhoods of points. Sequence Limits
35.2. Different types of sets 178
35.4. Multidimensional vector spaces 203
§ 36. Limit and continuity of functions of several variables
36.1. Functions of many variables 210
36.2. Displays. Display Limit 212
36.3. Continuity of mappings at a point 218
36.4. Display Limit Properties 220
36.5. Repeat Limits 221
36.6. Limit and continuity of the composition of mappings... 223
36.7. Continuous mappings of compacta 226
36.8. Uniform continuity 229
36.9. Continuous mappings of path-connected sets 233
36.10. Properties of continuous mappings 235
§ 37. Partial derivatives. Differentiability of functions of several variables 240
37.1. Partial derivatives and partial differentials ... . 240
37.2. Differentiability of functions at a point 244
37.3. Differentiation of a compound function 253
37.4. Invariance of the form of the first differential with respect to the choice of variables. Rules for Calculating Differentials 256
37.5. Geometric meaning of partial derivatives and total differential 262
37.6. Function Gradient 265
37.7. Directional derivative 265
37.8. An example of the study of functions of two variables .... 271

§ 38. Partial derivatives and differentials of higher orders 273
38.1. Partial derivatives of higher orders 273
38.2. Higher order differentials 277
§ 39. Taylor formula and Taylor series for functions of several variables 281
39.1. Taylor formula for functions of several variables. . 281
39.2. Finite Increment Formula for Functions of Many Variables 291
39.3. Estimation of the remainder term of the Taylor formula in the entire domain of the function 292
39.4. Uniform convergence with respect to the parameter of a family of functions 295
39.5. Remarks on Taylor Series for Functions of Several Variables 298
§ 40. Extrema of functions of several variables 299
40.1. Necessary Conditions for an Extremum 299
40.2. Sufficient conditions for a strict extremum 302
40.3. Remarks on extrema on sets 308
§ 41. Implicit functions. Displays 309
41.1. Implicit functions defined by a single equation. . 309
41.2. Set products 316
41.3. Implicit functions defined by a system of equations 317
41.4. Vector displays 328
41.5. Linear displays 329
41.6. Differentiable mappings 335
41.7. Mappings with non-zero Jacobian. Area conservation principle 344
41.8. Implicit functions defined by an equation in which the uniqueness conditions are violated. Singular points of plane curves 349
41.9. Variable substitution 360
§ 42. Dependence of functions 363
42.1. The concept of function dependency. Necessary Condition for Dependent Functions 363
42.2. Sufficient Conditions for the Dependence of Functions 365
§ 43. Conditional extremum 371
43.1. The concept of conditional extremum 371
43.2. Method of Lagrange multipliers for finding conditional extremum points 376
43.3*. Geometric interpretation of the Lagrange method 379
43.4*. Stationary points of the Lagrange function 381
43.5*. Sufficient conditions for conditional extremum points 388
Chapter 5
Integral calculus of functions of several variables
§ 44. Multiple integrals 393
44.1. The concept of volume in n-dimensional space (Jordan measure). Measurable sets 393
44.2. Sets of measure zero 414
44.3. Definition of a Multiple Integral 417
44.4. Existence of an integral 424
44.5*. On the integrability of discontinuous functions 431
44.6. Multiple Integral Properties 434
44.7*. Criteria for the integrability of the Riemann and Darboux functions
§ 45. Reduction of a multiple integral to an iterated one 451
45.1. Reduction of a double integral to an iterated one 451
45.2. Generalization to the n-dimensional case 459
45.3*. Generalized integral Minkowski inequality. . 462
45.4. Volume of the u-ball 464
45.5. Independence of measure from the choice of coordinate system... 465

45.6*. Newton-Leibniz and Taylor formulas 466
§ 46. Change of variables in multiple integrals 469
46.1. Linear mappings of measurable sets 469
46.2. Metric properties of differentiables
46.3. The formula for the change of variables in a multiple integral.. . 482
46.4. The geometric meaning of the absolute value of the Jacobian of the mapping 490
46.5. Curvilinear coordinates 491
§ 47. Curvilinear integrals 494
47.1. Curvilinear integrals of the first kind 494
47.2. Curvilinear integrals of the second kind 498
47.3. Valid Transformation Class Extension
47.4. Curvilinear integrals over piecewise smooth
47.5. Stieltjes integral 505
47.6*. Existence of the Stieltjes integral 507
47.7. Generalization of the concept of a curvilinear integral of the second kind 514
47.9. Calculating areas using curvilinear
47.10. The geometric meaning of the sign of the Jacobian of the mapping of a flat area 525
47.11. Conditions for the Independence of a Curvilinear Integral from the Integration Path 529
§ 48. Improper multiple integrals 539
48.1. Basic definitions 539
48.2. Improper Integrals of Nonnegative Functions 542
48.3. Improper integrals of functions,
§ 49. Some geometric and physical applications of multiple integrals 550
49.1. Calculating areas and volumes 550
49.2. Physical applications of multiple integrals 551
§ 50. Elements of the theory of surfaces 553
50.1. Vector Functions of Several Variables 553
50.2. Elementary surfaces 555
50.3. Equivalent elementary surfaces. Parametrically defined surfaces 557
50.4. Surfaces implicitly defined 567
50.5. Tangent Plane and Surface Normal 567
50.6. Explicit surface representations 574
50.7. The first quadratic form of the surface 578
50.8. Curves on a surface, calculating their lengths and angles between them 580
50.9. Surface area 581
50.10. Smooth Surface Orientation 584
50.11. Surface bonding 588
50.12. Orientable and non-orientable surfaces 592
50.13. Another approach to the concept of surface orientation... 593
50.14. Curvature of curves lying on a surface. The second quadratic form of the surface 598
50.15. Properties of the second quadratic surface form... 601
50.16. Planar Surface Sections 602
50.17. Normal surface sections 605
50.18. Principal curvatures. Euler formula 607
50.19. Calculating Principal Curvatures 611
50.20. Classification of surface points 613
§ 51. Surface integrals 617
51.1. Definition and properties of surface integrals... 617
51.2. Formula for Representing a Surface Integral of the Second Kind as a Double Integral 621
51.3. Surface integrals as limits of integral sums 623
51.4. Surface integrals over piecewise smooth surfaces 626
51.5. Generalization of the concept of a surface integral of the second kind 626
§ 52. Scalar and vector fields 631
52.2. On the invariance of the concepts of gradient, divergence
52.3. Gauss-Ostrogradsky formula. Geometric definition of divergence 640
52.4. Stokes formula. Geometric definition of a vortex. . 647
52.5. Solenoid vector fields 653
52.6. Potential vector fields 655
§ 53. Eigenintegrals depending on the parameter 663
53.1. Definition of integrals depending on the parameter; their continuity and integrability with respect to a parameter. . . 663
53.2. Differentiation of integrals depending
§ 54. Improper integrals depending on the parameter 668
54.1. Basic definitions. Uniform convergence of integrals depending on the parameter 668
54.2*. A criterion for the uniform convergence of integrals 674
54.3. Properties of improper integrals depending
54.4. Application of the theory of integrals depending on a parameter to the calculation of definite integrals 682
54.5. Euler integrals 686
54.6. Complex-Valued Functions of a Real Argument 691
54.7*. Asymptotic Behavior of the Gamma Function 694
54.8*. Asymptotic series 698
54.9*. Asymptotic expansion of the incomplete gamma function 702
54.10. Remarks on multiple integrals depending
Index 706
Index of basic symbols 713

Volume 3. CONTENTS
Chapter 7

Fourier series. Fourier integral
§ 55. Trigonometric Fourier series 4
55.1. Definition of the Fourier series. Statement of the main
55.2. The tendency of the Fourier coefficients to zero 10
55.3. Dirichlet integral. Localization principle 15
55.4. Convergence of Fourier series at point 19
55.5*. Convergence of Fourier series for functions satisfying the Hölder condition 31
55.6. Summation of Fourier series by the method of arithmetic means 34
55.7. Approximation of continuous functions by polynomials 40
55.8. Completeness of the trigonometric system and the system of non-negative integer powers x in the space of continuous functions 43
55.9. The minimal property of Fourier sums. Bessel's inequality and Parseval's equality 45
55.10. The nature of the convergence of Fourier series. Term Differentiation of Fourier Series 48
55.11. Term-by-Term Integration of Fourier Series 53
55.12. Fourier Series in the Case of an Arbitrary Interval 56
55.13. Complex notation of Fourier series 57
55.14. Expansion of a logarithm into a power series in the complex domain 58
55.15. Summation of trigonometric series 59
§ 56. Fourier integral and Fourier transform 61
56.1. Representation of functions as a Fourier integral 61
56.2. Different ways of writing the Fourier formula 70
56.3. Principal value of the integral 71
56.4. Complex notation of the Fourier integral 72
56.5. Fourier transform 73
56.6. Laplace integrals 76
56.7. Properties of the Fourier transform of absolutely integrable functions 77
56.8. Fourier transform of derivatives 78
56.9. Convolution and Fourier Transform 80
56.10. Derivative of the Fourier transform of a function 83
Chapter 8

functional spaces
§ 57. Metric spaces 85
57.1. Definitions and examples 85
57.2. Full spaces 91
57.3. Mappings of metric spaces 97
57.4. Contraction mapping principle 101
57.5. Completion of metric spaces 105
57.6. Compacts 110
57.7. Continuous mappings of sets 122
57.8. Connected sets 124
57.9. Arzel's criterion for the compactness of systems of functions 124
§ 58. Linear normed and semi-normed
58.1. Linear spaces 128
58.2. Norm and semi-norm 141
58.3. Examples of normalized and semi-normalized
58.4. Properties of semi-normed spaces 150
58.5. Properties of normed spaces 154
58.6. Linear operators 162
58.7. Bilinear mappings of normalized
58.8. Differentiable mappings of linear normed spaces 175
58.9. Finite Increment Formula 180
58.10. Higher order derivatives 182
58.11. Taylor Formula 184
§ 59. Linear spaces with inner product 186
59.1. Dot and almost dot product 186
59.2. Examples of Linear Spaces with Dot Product 191
59.3. Properties of linear spaces with scalar product. Hilbert spaces 193
59.4. factor-spaces 198
59.5. Space L2 202
59.6. Spaces Lp 214
§ 60. Orthonormal bases and expansions in them 217
60.1. Orthonormal systems 217
60.2. Orthogonalization 221
60.3. complete systems. Completeness of the trigonometric system and the system of Legendre polynomials 224
60.5. Existence of a basis in separable Hilbert spaces. Isomorphism of separable Hilbert spaces 239
60.6. Fourier series expansion of functions with square integrable 243
60.7. Orthogonal direct sum decompositions of Hilbert spaces 248
60.8. Functionals of Hilbert spaces 254
60.9*. Fourier transform of square-integrable functions. Plancherel's theorem 257
§ 61. Generalized functions 266
61.1. General considerations 266
61.2. Linear spaces with convergence. Functionals. Dual spaces 272
61.3. Definition of generalized functions. Spaces View" 277
61.4. Differentiation of generalized functions 283
61.5. The space of basic functions S and the space of generalized functions S" 287
61.6. Fourier transform in S 290 space
61.7. Fourier transform of generalized functions 293
Addition
§ 62. Some questions of approximate calculations 301
62.1. Application of the Taylor formula for the approximate calculation of the values of functions and integrals 301
62.2. Solving Equations 305
62.3. Function Interpolation 311
62.4. Quadrature formulas 314
62.5. Error of quadrature formulas 317
62.6. Approximate calculation of derivatives 321
§ 63. Partitioning a set into classes of equivalent elements 323
§ 64. Filter limit 325
64.1. Topological spaces 326
64.2. Filters 328
64.4. Display limit by filter 335
Subject index 340
Index of basic symbols 346

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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 given 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. The 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 (nonresonant and resonant cases). 9. Homogeneous system of linear ordinary differential equations (ODEs). Fundamental decision system and fundamental matrix. Vronskian. Liouville formula. Structure of the general solution of a 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. Conjugate, 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. The theorem on the infinity of 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., 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, 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 of 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 of 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 in 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 of educational activity of 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. Methods for studying the topic "Inverse trigonometric functions." 41. Methods for studying the topic "General methods for solving equations in the school mathematics course." 42. Methods for studying the topic "Quadricular 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. A school course in 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: 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. A necessary condition for a conditional extremum. Method of Lagrange multipliers. 27. Number series. Cauchy Convergence Criterion

Ministry of Education and Science of the Russian Federation Federal State Budgetary Educational Institution of Higher Professional 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"

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

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

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

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I APPROVE Department of Physical and Mathematical Disciplines E.N.

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 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 ANNOTATION OF THE EDUCATIONAL DISCIPLINE Direction of training 10.03.01 "Information security" orientation (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

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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 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 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 the set of 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 on the limit point (Bolzano-Weierstrass principle).
- § 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 meaning of the 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 atomic 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. The main general methods of finding the antiderivative.
  - 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 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. 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

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Chapter IX. Continuous mappings (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 of a compact (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 of 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 higher-order differentials (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 Stokes formula (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 ideas about 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 related to the concept of a Fourier series
1. Orthogonal systems of functions (488).
2. Fourier Coefficients 494
3. Fourier series 499
4. On one important source of 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 about 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.