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

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

T. 3. Harmonic analysis. Elements of functional analysis.

M .: Bustard; vol. 1- 2003, 704s.; vol. 2- 2004, 720s .; vol. 3- 2006, 351s.

The tutorial matches 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 university students of both physics and mathematics, and engineering and physical specialties of technical colleges, as well as students of other specialties for in-depth mathematical training.

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Volume 1. Table of contents
Foreword 3
Introduction 7
Chapter 1
Differential calculus of functions of one variable
§ 1. Sets and functions. Logic symbols 13
1.1. Sets. Set operations 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 real numbers 35
2.2 *. Addition and Multiplication Properties 39
2.3 *. Ordering properties 47
2.4 *. Continuity property of real numbers 51
2.5 *. Sections in the set of real numbers 52
2.6*. Rational degrees real numbers 58
2.7. Binomial Newton formula 60

§ 3. Number sets 63
3.1. Extended number line 63
3.2. Real number spans. Neighborhood 64
3.3. Limited and Unlimited Sets 68
3.4. Upper and lower bounds of number sets 70
3.5 *. Top and Bottom Arithmetic Properties ... 75
3.6. Archimedes principle 78
3.7. Nested Line Principle 80
3.8 *. Uniqueness of a continuous ordered field ... 85
§ 4. Limit numerical sequence 92
4.1. Determining the Limit of a Number Sequence 92
4.2. Uniqueness of the Limit of a Number Sequence ... 100
4.3. Passing to the Limit in Inequalities 101
4.4. Boundedness of Converging Sequences 107
4.5. Monotone sequences 108
4.6. Bolzano-Weierstrass theorem 113
4.7. Cauchy's criterion for the convergence of a sequence 115
4.8. Infinitesimal Sequences 118
4.9. Limit Properties Associated with Arithmetic Operations on Sequences 120
4.10. The representation of real numbers is infinite decimal fractions 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. Methods for Setting Functions 156
5.3. Elementary functions and their classification 160
5.4. First definition of the limit of a function 162
5.5. Continuous Functions 172
5.6. The condition for the existence of the limit of a function 177
5.7. Second Definition of Function Limit 179
5.8. Limit of a function on the union of sets 184
5.9. One-Sided Limits and One-Sided Continuity ... 185
5.10. Function Limit Properties 189
5.11. Infinitesimal and Infinitely Large Functions 194
5.12. Various shapes continuity records
5.13. Function breakpoint classification 202
5.14. Limits of Monotone Functions 204
5.15. Cauchy criterion for the existence of the limit of a function 210
5.16. The 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 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. Feature Comparison 253
8.3. Equivalent Functions 264
8.4. Method of highlighting the main part of a function and its application to the calculation of limits 267
§ 9. Derivative and differential 271
9.1. Definition of the derivative 271
9.2. Differential function 274
9.3. Geometric meaning derivative and differential ... 280
9.4. Physical sense derivative and differential 284
9.5. Derivative Computation Rules Related to Arithmetic Operations on Functions 288
9.6. Derivative of the inverse function 291
9.7. Derivative and Differential of a Composite Function 294
9.8. Hyperbolic functions and their derivatives 301
§ten. Higher-order derivatives and differentials 304
10.1. Higher-order derivatives 304
10.2. Higher-order derivatives of the sum and products of functions 306
10.3. Higher order derivatives of complex functions, from inverse functions and from the functions given
10.4. Higher-order differentials 311
§eleven. Mean value theorems for differentiable functions 313
11.1 Fermat's theorem

11.2. Rolle's, Lagrange's and Cauchy's theorems on mean values. ... 316
§12. Disclosure of Uncertainties under L'Hôpital's Rule 327
12.1 Uncertainties of the form 0/0
12.2 Uncertainties of a kind ----

12.3. Generalization of L'Hôpital's rule 337
§ 13. Taylor's formula 339
13.1. Derivation of Taylor's formula 339
13.2. Taylor polynomial as a polynomial of best approximation of a function in a neighborhood of a given point 344
13.3. Taylor's formulas for basic elementary
13.4. Calculating Limits Using Taylor's Formula (Principal Partition Method) 351
§ 14. Investigation of the behavior of functions 353
14.1. Monotonicity of a function 353
14.2. Finding the greatest and 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. Length of Curve 397
16.3. Curve orientation. Arc of a curve. The sum of the curves. Implicit Curves 408
16.4. The tangent to the 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 Evolution of Curve 436
17.5. Formulas for curvature and evolute of a plane curve ... 437
17.6. Evolvent 444
17.7. Spatial Curve Torsion 447
17.9. Torsion Formulas 451
Chapter 2
Integral calculus of functions of one variable
§eighteen. Definitions and properties of an indefinite integral 453
18.1. Antiderivative and Indefinite Integral 453
18.2. Basic properties of the integral 456
18.3. Tabular Integrals 458
18.4. Integration by substitution (variable substitution) 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 complex numbers 481
19.3. Some Concepts of Analysis in the Field of Complex Numbers 482
19.4. Factoring Polynomials 486
19.5 *. The greatest common divisor polynomials 490
19.6. Decomposition of regular rational fractions into elementary ones 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 \ Ux, 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. Integrals types 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 expressed in terms of elementary functions 532
§ 23. The definite integral 533
23.1. Definition of the Riemann integral 533
23.2 *. Cauchy's 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 *. Criteria for the integrability of Darboux and Riemann 551
23.8 *. Fluctuation of functions 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. The first mean value theorem for a definite integral 583
§25. Definite integral with variable limits
25.1. Integral Continuity Along the Upper Limit
25.2. Differentiability of the integral over the upper limit of integration. The existence of an antiderivative for a continuous function 588
25.3. Newton-Leibniz formula 591
25.4 *. The existence of a generalized antiderivative. Newton-Leibniz formula for generalized antiderivative. ... 592
§26. Variable Change Formulas in Integral and Integration by Parts 596
26.1. Variable Replacement 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. Properties of the measure of open sets 612
§28. Some geometric and physical applications of the definite integral 618
28.1. Area Calculation 618
28.2 *. Integral Hölder and Minkowski Inequalities ... 625
28.3. The volume of a body of revolution 630
28.4. Calculating the Length of a Curve 632
28.5. Surface of revolution 637
28.6. Work of power 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's 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
Subject and nominal index 685
Index to basic symbols 695

Volume 2. Table of contents
Foreword 3
Chapter 3

Rows
§ 30. Number series 5
30.1. Definition of a series and its convergence 5
30.2. Properties of converging series 9
30.3. Cauchy Criterion for Convergence of Series 11
30.4. Rows with non-negative members 13
30.5. Comparison criterion for series with non-negative members. Method of highlighting 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 convergence of series with nonnegative terms 23
30.8 *. Hölder and Minkowski inequalities for finite and infinite sums 25
30.9. Alternating rows 27
30.10. Rows converging absolutely. Application of absolutely convergent series to the study of convergence
30.11. D'Alembert and Cauchy Tests for Arbitrary Number Series 38
30.12. Converging rows that do not converge absolutely. Riemann's Theorem 39
30.13. Abel transform. Convergence criteria for Dirichlet and Abel 43
30.14 *. Asymptotic behavior of the remainders of convergent series and partial sums of divergent series 48
30.15. Summability of series by the method of arithmetic means 52
§ 31. Endless works 53
31.1. Basic definitions. The Simplest Properties of Infinite Products 53
31.2. Cauchy's criterion for convergence of infinite products 57
31.3. Endless works with valid
31.4. Absolutely converging endless works ... 62
31.5 *. The Riemann Zeta Function and 65 Prime Numbers
§ 32. Functional sequences and series 67
32.1. Convergence of functional sequences
32.2. Uniform convergence of functional sequences 71
32.3. Uniformly converging functional series 79
32.4. Properties of Uniformly Convergent Series and Sequences 90
§ 33. Power series 100
33.1. Convergence radius and circle of convergence of a power series 100
33.2 *. Cauchy-Hadamard formula for the radius of convergence
33.3. Analytical functions 110
33.4. Analytical Functions in Real Domain ... 112
33.5. Expansion of functions in power series. Different ways to write the remainder of the Taylor formula. ... 116
33.6. Expansion of elementary functions in a Taylor series ... 121
33.7. Methods for Expanding Functions in Power Series 131
33.8. Sterling's 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 Functional Series 162
Chapter 4
Differential calculus of functions of several variables
§ 35. Multidimensional spaces 165
35.1. Vicinity 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. Mappings. Limit of mappings 212
36.3. Continuity of mappings at point 218
36.4. Display Limit Properties 220
36.5. Repeated limits 221
36.6. The Limit and Continuity of the Composition of Mappings ... 223
36.7. Continuous Mappings of Compact Sets 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 point 244
37.3. Differentiating a Complex Function 253
37.4. Invariance of the form of the first differential with respect to the choice of variables. Differential Computation Rules 256
37.5. The geometric meaning of partial derivatives and total differential 262
37.6. Gradient function 265
37.7. Directional derivative 265
37.8. An example of studying 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's formula and Taylor series for functions of several variables 281
39.1. Taylor's formula for functions of several variables. ... 281
39.2. Formula of finite increments for functions of several variables 291
39.3. Estimate of the remainder of the Taylor formula in the entire domain of definition of the function 292
39.4. Uniform Convergence in Parameter of a Family of Functions 295
39.5. Remarks on Taylor Series for Functions of Several Variables 298
§ 40. Extremums 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. Views 309
41.1. Implicit functions defined by one equation. ... 309
41.2. Set Art 316
41.3. Implicit Functions Defined by a System of Equations 317
41.4. Vector mappings 328
41.5. Linear Mappings 329
41.6. Differentiable Mappings 335
41.7. Mappings with non-zero Jacobian. Area preservation 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. Replacing Variables 360
§ 42. Dependence of functions 363
42.1. Function dependency concept. A necessary condition for the dependence of functions 363
42.2. Sufficient conditions for the dependence of functions 365
§ 43. Conditional extremum 371
43.1. The concept of a conditional extremum 371
43.2. Lagrange multiplier method for finding the points of conditional extremum 376
43.3 *. Geometric interpretation of the Lagrange method 379
43.4 *. Stationary points of the Lagrange function 381
43.5 *. Sufficient conditions for the points of conditional extremum 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 multiple integrals 417
44.4. The existence of an integral 424
44.5 *. On the integrability of discontinuous functions 431
44.6. Properties of the multiple integral 434
44.7 *. Criteria for the integrability of the Riemann and Darboux functions
§ 45. Reduction of a multiple integral to a repeated one 451
45.1. Reduction of a Double Integral to a Repeated Integral 451
45.2. Generalization to the n-dimensional case 459
45.3 *. Generalized integral Minkowski inequality. ... 462
45.4. Volume of i-dimensional ball 464
45.5. Independence of the Measure from the Choice of the 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 differentiable
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. Extending the class of admissible transformations
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 curved lines
47.10. The geometric meaning of the sign of the Jacobian of the flat-area mapping 525
47.11. Conditions for the independence of the curvilinear integral from the path of integration 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. Calculation of 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. Implicit surfaces 567
50.5. Tangent Plane and Normal to Surface 567
50.6. Explicit Surface Representations 574
50.7. The first quadratic form of the surface 578
50.8. Curves on the 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. Oriented and non-oriented surfaces 592
50.13. Another Approach to Surface Orientation ... 593
50.14. Curvature of curves lying on a surface. Second quadratic form of surface 598
50.15. Properties of the second quadratic surface shape ... 601
50.16. Flat surface sections 602
50.17. Normal surface sections 605
50.18. Principal curvatures. Euler's formula 607
50.19. Calculation of Principal Curvatures 611
50.20. Surface point classification 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. Solenoidal Vector Fields 653
52.6. Potential Vector Fields 655
§ 53. Proper integrals depending on a parameter 663
53.1. Determination of integrals depending on a parameter; their continuity and integrability with respect to a parameter. ... ... 663
53.2. Differentiation of integrals depending
§ 54. Improper integrals depending on a parameter 668
54.1. Basic definitions. Uniform convergence of integrals depending on a parameter 668
54.2 *. A criterion for the uniform convergence of integrals 674
54.3. Properties of improper integrals depending on
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 Real Argument 691
54.7 *. Asymptotic Behavior of the Gamma Function 694
54.8 *. Asymptotic series 698
54.9 *. Asymptotic expansion of an incomplete gamma function 702
54.10. Remarks on multiple integrals depending
Subject-nominal index 706
Index to Basic Symbols 713

Volume 3. CONTENTS
Chapter 7

Fourier series. Fourier integral
§ 55. Trigonometric Fourier series 4
55.1. Determination 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 a trigonometric system and a system of nonnegative integer powers of x in the space of continuous functions 43
55.9. Minimal property of Fourier sums. Bessel's inequality and Parseval's equality 45
55.10. The nature of the convergence of the Fourier series. Term-by-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 the logarithm in a power series in the complex domain 58
55.15. Summation of trigonometric series 59
§ 56. The Fourier integral and the Fourier transform 61
56.1. Representation of functions in the form of a Fourier integral 61
56.2. Different types of writing the Fourier formula 70
56.3. The main 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 function 83
Chapter 8

Functional spaces
§ 57. Metric spaces 85
57.1. Definitions and Examples 85
57.2. Full spaces 91
57.3. Metric Space Mappings 97
57.4. The Contraction Mapping Principle 101
57.5. Completion of metric spaces 105
57.6. Compact 110
57.7. Continuous Mappings of Sets 122
57.8. Connected sets 124
57.9. Arzela's criterion for compactness of systems of functions 124
§ 58. Linear normalized and seminormalized
58.1. Linear spaces 128
58.2. Norm and Seminorm 141
58.3. Examples of normalized and semi-normalized
58.4. Properties of semi-normalized 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 normed linear spaces 175
58.9. End increment formula 180
58.10. Higher order derivatives 182
58.11. Taylor's Formula 184
§ 59. Linear spaces with scalar product 186
59.1. Scalar and Near-Scalar Products 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-space 198
59.5. Space L2 202
59.6. Lp spaces 214
§ 60. Orthonormal bases and their expansions 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. Expansion of functions with square integrable in a Fourier series 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. Associated spaces 272
61.3. Definition of generic functions. View spaces "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 space S 290
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 Computation 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. Filter Display Limit 335
Subject and nominal index 340
Index to basic symbols 346

Transcript

2 Mathematical analysis 1. Completeness: supremum and infimum of a numerical set. The principle of nested segments. Irrationality of the number Theorem on the existence of the limit of a monotone sequence. The number e. 3. Equivalence of definitions of the limit of a function at a point in the language and in the language of sequences. Two wonderful limits. 4. Continuity of a function of one variable at a point, a point of discontinuity and their classification. Properties of a function that is continuous on a segment. 5. Weierstrass theorems on the largest and the smallest values ​​of a continuous function defined on a segment. 6. Uniformity of continuity. Cantor's theorem. 7. The concept of the derivative and differentiability of a function of one variable, the differentiation of a complex function. 8. Derivatives and differentials of higher orders of functions of one variable. 9. Study of a function using derivatives (monotonicity, extrema, convexity and inflection points, asymptotes). 10. Parametrically specified functions and their differentiation. 11. Rolle's, Lagrange's and Cauchy's theorems. 12. L'Hôpital's rule. 13. Taylor's formula with a remainder in the Lagrange form. 14. Local Taylor formula with remainder in the Peano form. Expansion of basic elementary functions by Taylor's formula. 15. A criterion for the Riemann integrability of 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 indefinite integral. Integration of rational fractions. 18. Methods for the approximate calculation of definite integrals: methods of rectangles, trapezoids, parabolas. 19. A definite integral with a 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 I and II kind. Convergence criteria. 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. Lagrange multiplier method. 27. Number series. Cauchy's criterion for convergence of a series. 28. Cauchy test for convergence of positive series 29. D'Alembert test for convergence of positive series 30. Leibniz's theorem on the convergence of alternating series. 31. Cauchy's criterion for uniform convergence of function series. 32. Sufficient conditions for the 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 set of convergence of a power series.

3 34. Multiple Riemann integral, its existence. 35. Reduction of a multiple integral to a repeated one. References 1. Kartashev, A.P. Mathematical analysis: textbook. - 2nd ed., Stereotype. - SPb .: Lan, p. 2. Kirkinskiy, A.S. Mathematical analysis: a textbook for universities. - M .: Academic Project, p. 3. Kudryavtsev, L. D. A short course in mathematical analysis. Vol. 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.Sadovnichy. - M .: Research Center "RHD", Nikolsky, S.M. The course of mathematical analysis. T. 1, 2.- Ed. 4th, rev. and additional - Moscow: Science, p. 6. Ilyin, V.A. Fundamentals of mathematical analysis. Part 1, 2. - Ed. 4th, rev. and additional - Moscow: Science, p. Differential Equations. 1. An existence and uniqueness theorem for the solution of the Cauchy problem for an ordinary differential equation of the first order. 2. The theorem of existence and uniqueness of the solution of the Cauchy problem for an ordinary differential equation of the first order 3. The theorem on the continuous dependence of the solution of the Cauchy problem for an ordinary differential equation of the first order on the parameters and on the initial data. 4. The theorem on the differentiability of 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 (ODE). General properties. Homogeneous ODE. Fundamental decision system. Vronskian. Liouville's 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 decision system. 8. Inhomogeneous linear ordinary differential equations with constant coefficients with inhomogeneity in the form of a quasipolynomial (nonresonant and resonant cases). 9. Homogeneous system of linear ordinary differential equations (ODE). Fundamental decision system and fundamental matrix. Vronskian. Liouville's formula. The structure of a general solution to 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 decision system. 12. Inhomogeneous system of ordinary differential equations with constant coefficients with inhomogeneity in the form of a matrix with elements of quasipolynomials (nonresonant 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 the solution of a boundary value problem. 14. Autonomous systems. Properties of solutions. Singular points of a linear autonomous system of two equations. Stability and asymptotic stability according to Lyapunov. Stability of a homogeneous system of linear differential equations with a variable matrix. 15. 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., Rev. - M .: Publishing house of the Bauman Moscow State Technical University, p. 3. Egorov, A.I. Ordinary differential equations with applications - Ed. 2nd, revised - 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, p. - Moscow: Fizmatlit, p. 6. Phillips, G. Differential equations: translation from English / G. Phillips; edited by A.Ya. Khinchin. - 4th ed., P. - Moscow: KomBook, p. Algebra and number theory 1. Definition of a group, ring and field. Examples. Construction of the field of complex numbers. Exponentiation of complex numbers. Extracting the root of 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). Study of SLAE. Gauss method. Cramer's rule. 5. The ring of polynomials in one variable. Division theorem with remainder. GCD of two polynomials. 6. Roots and multiple roots of a polynomial. The main 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. The rank of the matrix. SLAE compatibility. 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 complement. 13. Linear operators in linear spaces and operations on them. Linear operator matrix. Linear operator matrices in various bases.

5 14. Image and kernel, rank and defect of a linear operator. Kernel and image dimensions. 15. Invariant subspaces of a linear operator. Eigenvectors and eigenvalues ​​of a linear operator. 16. A criterion for 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 forms of quadratic forms. 20. Sign-constant 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. Continuous (continued) fractions. Suitable fractions. 23. Prime numbers. Sieve of Eratosthenes. The theorem on the infinity of primes. Decomposition of a number into prime factors 24. Antje function. Multiplicative function. Mobius function. Euler's function. 25. Comparisons. Basic properties. Complete deduction system. Reduced 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 in a compound module. 28. Comparisons of the second degree. Legendre symbol. 29. Primitive roots. 30. Indices. Applying indices to solving comparisons. References 1. Kurosh, A.G. Lectures on general algebra: textbook / A.G. Kurosh. - 2nd ed., Ster. - SPb .: Publishing house "Lan", p. 2. Birkhoff, G. Modern applied algebra: textbook / Garrett Birkhoff, Thomas K. Barty; translation from English by Yu.I. Manin. - 2nd ed., P. - St. Petersburg: Lan, p. 3. Ilyin, V.A. Linear Algebra: a textbook for students of physical specialties and the specialty "Applied Mathematics". - Ed. 5th, ster. - Moscow: FIZMATLIT, Kostrikin, A.I. Introduction to algebra. Part 1. Fundamentals of algebra: a textbook for university students studying in the specialties "Mathematics" and "Applied Mathematics" .- Ed. 2nd, rev. - Moscow: FIZMATLIT, Vinogradov, I.M. Fundamentals of number theory: textbook. - Ed. 11th - St. Petersburg; Moscow; Krasnodar: Lan, 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 in a plane defined in various ways. The relative position of two straight lines. The angle between two straight lines. 3. Transformation of coordinates when passing 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 given in different ways. The relative position of the two planes. Distance from point to plane. The angle between two planes. 8. Equations of a straight line in space. The relative position of two straight lines, a straight line and a plane. Distance from point to line. The angle between two straight lines, a straight 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. Methods for defining a curve. Curve length (definition and calculation). 12. Curvature and torsion of a curve. 13. Accompanying benchmark of a smooth curve. Freinet's formulas. 14. The first quadratic form of a smooth surface and its applications. 15. Second quadratic form of a smooth surface, normal curvature of the surface. 16. Principal directions and principal surface curvatures. 17. Lines of curvature and asymptotic lines of the surface. 18. Average and Gaussian curvature of the surface. 19. Topological space. Continuous displays. Homeomorphisms. Examples. 20. Euler's characteristic of a manifold. Examples. Literature 1. Nemchenko, K.E. Analytical geometry: textbook. - Moscow: Eksmo, p. 2. Dubrovin, B.A. Modern geometry: methods and applications. Vol. 1, 2. Geometry and topology of manifolds. - 5th ed. rev. - Moscow: Editorial URSS, p. 3. Zhafyarov, A. Zh. Geometry. In 2 hours a textbook. - 2nd ed. - Novosibirsk: Siberian University Publishing House, p. 4. Efimov, N.V. A short course in analytical geometry: a textbook for students of higher educational institutions. - 13th ed. - Moscow: FIZMATLIT, p. 5. Taimanov, I.A. Lectures on differential geometry. - Moscow; Izhevsk: Institute for Computer Research, p. 6. Atanasyan L.S., Bazyrev V.T. Geometry, part 1.2. Moscow: Knorus, p. 7. Rashefsky P.S. Differential geometry course. Moscow: Science, p. Theory and methods of teaching mathematics 1. The content of teaching mathematics in secondary school. 2. Didactic principles of teaching mathematics. 3. Methods of scientific knowledge. 4. Visibility in teaching mathematics. 5. Forms, methods and means of control and assessment of students' knowledge and skills. Norms of marks. 6. Extracurricular activities in mathematics. 7. Mathematical concepts and methods of their formation. 8. Problems as a means of teaching mathematics. 9. In-depth study of mathematics: content, methods and forms of organization of training. 10. Types of mathematical judgments: axiom, postulate, theorem.

7 11. Summary of a lesson in mathematics. 12. A lesson in mathematics. Types of lessons. Lesson analysis. 13. Studying mathematics in a small school: content, methods and forms of organization of training. 14. New learning technologies. 15. Differentiation of teaching mathematics. 16. Individualization of teaching mathematics. 17. Motivation of the educational activity of schoolchildren. 18. Logical and didactic analysis of the topic. 19. Technological approach to teaching mathematics 20. Humanization and humanization of teaching mathematics. 21. Education in the process of teaching mathematics. 22. Methodology for studying identical transformations. 23. Methodology for studying inequalities. 24. Methodology for studying the function. 25. Methodology for studying the topic "Equations and inequalities with a modulus". 26. Methods for studying the topic "Cartesian coordinates". 27. Methods for studying polyhedra and round bodies. 28. Methods for studying the topic "Vectors". 29. Technique for solving movement problems. 30. Methodology for solving problems for joint work. 31. Methodology for studying the theme "Triangles" 32. Methodology for studying the theme "Circle and circle". 33. Technique 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. Methodology for studying the topic "Solving equations and inequalities with parameters." 37. Methods for 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. Methodology for studying the topic "General methods for solving equations in the school course of mathematics." 42. Methods 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 "Using the derivative in the study of functions" References 1. Argunov, BI. School course in mathematics and methods of teaching it. - Moscow: Education, p. 2. Zemlyakov, A.N. Geometry in the 11th grade: guidelines for the textbook. A.V. Pogorelova: a guide for teachers. - 3rd ed., Door. - 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 et al. - 2nd ed. - M .: Education, p. 4. Latyshev, L.K. Translation: theory, practice and teaching methods: textbook. - 3rd ed., P. - Moscow: Academy, p. 5. Methods and technology of teaching mathematics: a course of lectures: a textbook for students of mathematics faculties of higher educational institutions studying in the direction (050200) physics and mathematics education. - Moscow: Bustard, p.

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

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

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NEW. Natanzon S.M. A short course in mathematical analysis. 2004 year. 98 pages djvu. 1.2 Mb.
This publication is a summary of the course of lectures given by the author for 1st year students of the Independent Moscow University in 1997-1998 and 2002-2003 academic years.

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NEW. E.B. Boronin. Mathematical analysis. Lecture notes. 2007 year. 160 pages pdf. 2.1 Mb.
This book is written for technical students who wish to prepare for the mathematical analysis exam. The content of this book is fully consistent with the program for the course "Mathematical Analysis", the exam for which is provided in most higher educational institutions in Russia. The program helps to quickly and without unnecessary difficulties find the necessary answer to the question posed.
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Arkhipov, Sadovnichy, Chubarikov. Lectures on mathematical analysis. Tutorial analysis. 1999 year. 635 pp. Djvu. 5.2 MB.
The book is a textbook on the course of mathematical analysis and is devoted to differential and integral calculus of functions of one and several variables. It is based on the lectures given by the authors at the Faculty of Mechanics and Mathematics of Moscow State University. M.V. Lomonosov. The textbook proposes a new approach to the presentation of a number of basic concepts and theorems of analysis, as well as to the content of the course itself. For students of universities, pedagogical universities and universities with in-depth study of mathematics

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A.P. Aksenov Mathematical analysis. (Fourier series. Fourier integral. Summation of divergent series.) Textbook. 1999 year. 86 pages PDF 1.2 Mb.
The manual corresponds to the state standard of the discipline "Mathematical Analysis" of the direction of bachelor's training 510200 "Applied Mathematics and Informatics".
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Aksenov. Mathematical analysis. (Integrals depending on a parameter. Double integrals. Curvilinear integrals.) Textbook SPb. year 2000. 145 pp. PDF. Size 2.3 Mb. djvu.
The manual corresponds to the state standard of the discipline "Mathematical Analysis" of the direction of bachelor's training 510200 "Applied Mathematics and Informatics". Contains a presentation of theoretical material in accordance with the current program on the topics: "Integrals depending on a parameter, proper and improper", "Double integral", "Curvilinear integrals of the first and second kind", "Calculation of the areas of curved surfaces, given both explicit and parametric equations "," Euler integrals (Beta function and Gamma function) ". Disassembled a large number of examples and problems (a total of 47).

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De Bruyne. Asymptotic methods in analysis. 245 pages djvu. 1.6 Mb.
The book contains an elementary presentation of a number of methods used in analysis to obtain asymptotic formulas. The importance of the methods described in the book, the clarity and accessibility of the presentation make this book very valuable for all beginners to get acquainted with such methods. The book is of undoubted interest also for those who are already familiar with this area of ​​analysis.

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Stefan Banach. Differential and integral calculus. 1966 year. 437 pp. Djvu. 7.7 Mb.
Stefan Banach is one of the greatest mathematicians of the 20th century. This book was conceived by him as a manual for an initial acquaintance with the subject. Meanwhile, the author managed to masterfully illuminate almost all the main material of differential and integral calculus in a small volume book, without frightening off the reader with scrupulous rigor of presentation.
The book is distinguished by its simplicity and brevity of presentation. It contains many well-chosen examples, as well as problems for independent solution. Designed for students of technical colleges (especially correspondence courses), pedagogical institutes, as well as for engineering and technical workers who wish to refresh their memory on the basic facts of differential and integral calculus.
When preparing the second edition, the experience of teaching this book in some higher technical educational institutions was taken into account; in this regard, a small number of additions have been made to the book, and some places in the text have been corrected. This brought the book closer to the level of modern textbooks on mathematical analysis and made it possible to use it in technical colleges.

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B.M. Budak, S.V. Fomin. Multiples and ranks. Textbook. 1965. 606 pp. Djvu. 4.6 Mb.
For phys.-mat. faculties of universities.
RECOMMEND!!!. Especially for PHYSICISTS.

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Viosagmir I.A. Higher mathematics for dummies. Function limit. 2011. 95 pp. Pdf. 6.1 Mb.
I welcome you to my first book on function limits. This is the first part of my future series "Higher Math for Dummies". The title of the book should already tell you a lot about it, but you may not understand it at all. This book is not dedicated to “dummies”, but to all those who find it difficult to understand what professors do in their books. I am sure that you understand me. I myself have been and am in such a situation that I simply have to read the same sentence several times. This is fine? I think no.
So how is my book different from all the others? Firstly, this is a normal language, not “abstruse”; secondly, a lot of examples have been analyzed here, which, by the way, will certainly be useful to you; thirdly, the text has a significant difference between itself - the main things are highlighted with certain markers, and finally, my goal is only one - your understanding. Only one thing is required of you: desire and skill. "Skills?" - you ask. Yes! Ability to remember and understand.

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V.N. Gorbuzov. Mathematical analysis: integrals depending on parameters. Uch. allowance. 2006 year. 496 pages PDF. 1.6 Mb.
Differential and integral calculus of functions given by certain improper integrals that depend on parameters are presented. It is intended for university students studying in mathematics and physics specialties, as well as for students of technical specialties with an extended program in mathematics.

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Dorogovtsev A.Ya. Mathematical analysis. A short course in a modern way. Second edition. 2004 year. 560 pages djvu. 5.1 MB.
The book contains a short and at the same time quite complete in terms of coverage of the material presentation of the modern course of mathematical analysis. The book is designed primarily for students of universities and technical universities and is intended for the initial study of the course. A modernized presentation of a number of sections is given: functions of several variables, multiple integrals, integrals over manifolds, an explanation of the Stokes formula, etc. The theoretical material is illustrated by a large number of exercises and examples. ... For university students, teachers of mathematics, engineering and technical workers.

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Egorov V.I., Salimova A.F. Definite and multiple integrals. Elements of field theory. 2004 year. 256 pages djvu. 1.6 Mb.
The publication presents the theory and main applications of definite and multiple integrals, as well as elements of field theory. The material is adapted to the modern program of mathematical education in higher technical educational institutions, for use in computer training systems. The book is intended for students of technical universities. It can also be useful for teachers, engineers, scientists.
Clearly printed book. All statements of the theory are shown by examples. I recommend it as additional literature for understanding the material.

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Evgrafov. Asymptotic estimates and entire functions. 320 pages djvu. 3.2 Mb.
The book is devoted to the presentation of various methods of asymptotic estimates (Laplace's method, saddle point method, theory of residues) used in the theory of entire functions. The methods are illustrated mainly on the basis of this theory. The main facts from the theory of entire functions are not supposed to be known to the reader - their presentation is organically included in the structure of the book. A chapter on the asymptotics of conformal mappings has been added to the 3rd edition. The book is designed for a wide contingent of readers - from students to scientific workers, both mathematicians and applied specialists.

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I WOULD. Zel'dovich, I.M. Yaglom. Higher mathematics for novice physicists and technicians. 1982 year. 514 pp. Djvu. 12.3 Mb.
This book is an introduction to calculus. Along with the presentation of the beginnings of analytical geometry and mathematical analysis (differential and integral calculus), the book contains the concepts of power and trigonometric series and the simplest differential equations, and also touches on a number of sections and topics from physics (mechanics and theory of oscillations, theory of electrical circuits, radioactive decay , lasers, etc.). The book is intended for readers interested in the natural science applications of higher mathematics, teachers of universities and technical colleges, as well as future physicists and engineers.

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Zeldovich, Yaglom. The book is in three parts: 1. Elements of higher mathematics. Contains: Functions and graphs (50 pages) (, What is a derivative (50 pages), What is an integral (20 pages), Calculation of derivatives (20 pages), Integration techniques (20 pages), Series, elementary differential equations (35 pages), Investigation of functions, several problems in geometry (55 pages). 2. Applications of higher mathematics to some questions of physics and technology (160 pages). Contains: Radioactive decay and fission of nuclei, Mechanics, Oscillations, Thermal motion of molecules, distribution of air density in the atmosphere, Absorption and emission of light, lasers, Electric circuits and oscillatory movements in them 3. Additional topics from higher mathematics (50 pages). Contains: Complex numbers, What functions do physics need, Wonderful Dirac delta function, Some applications of the complex variable function and delta functions 4. Appendices, Answers, Directions, Solutions. Truncated what kind of book? You can go crazy by reading one table of contents. But this is not a textbook on mathematics, THIS BOOK IS ABOUT HOW TO USE MATHEMATICS. By the way, studying it, you will inevitably learn physics as well. Super. djvu, 500 pages. Size 8.7 MB.

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Zorich V.A. Mathematical analysis. In 2 parts. Textbook. 1 - 1997, 2 - 1984. 567 + 640 pages djvu. 9.6 + 7.4 Mb.
University textbook for students of physical and mathematical specialties. It can be useful for students of faculties and universities with advanced mathematical training, as well as specialists in mathematics and its applications. The book reflects the connection between the course of classical analysis and modern mathematical courses (algebra, differential geometry, differential equations, complex and functional analysis).
The first part included: an introduction to analysis (logical symbols, 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 tutorial includes the following sections: Multivariate integral. Differential forms and their integration. Series and integrals depending on a parameter (including series and Fourier transforms, as well as asymptotic expansions).

Problem solving manuals.

NEW. Sadovnnichaya I.V., Khoroshilova E.V. The definite integral: theory and practice of computation. 2008 year. 528 pp. Djvu. 2.7 Mb.
The publication is devoted to the theoretical and practical aspects of calculating definite integrals, as well as methods of their estimation, properties and applications to the solution of various geometric and physical problems. The book contains sections devoted to methods for calculating proper integrals, properties of improper integrals, geometric and physical applications of a definite integral, as well as some generalizations of the Riemann integral - Lebesgue and Stieltjes integrals.
The presentation of the theoretical material is supported by a large number (more than 220) analyzed examples of the calculation, estimation and study of the properties of definite integrals; at the end of each paragraph, there are tasks for independent solution (more than 640, the overwhelming majority - with solutions).
The purpose of the manual is to help the student during the passage of the topic "The definite integral" in lectures and practical lessons. A student can contact him to obtain background information on the issue that has arisen. The book can also be useful for teachers and everyone who wants to study this topic in sufficient detail and widely.

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NEW. Khoroshilova E.V. Mathematical analysis: indefinite integral. (to help practical exercises). 2007 year. 184 pp. Djvu. 822 Kb.
The book provides basic theoretical information about indefinite integrals, considers most of the known techniques and methods of integration and various classes of integrable functions (indicating the methods of integration). The presentation of the material is supported by a large number of analyzed examples of calculating integrals (more than 200 integrals), at the end of each paragraph there are problems for independent solution (more than 200 problems with answers).
The manual contains the following paragraphs: "The concept of an indefinite integral", "Basic methods of integration", "Integration of rational fractions", "Integration of irrational functions", "Integration of trigonometric functions", "Integration of hyperbolic, exponential, logarithmic and other transcendental functions." The book is intended for mastering the theory of the indefinite integral in practice, developing the skills of practical integration, consolidating the course of lectures, using it at seminars and during the preparation of homework. The purpose of the manual is to help the student in mastering various techniques and methods of integration.
For university students, including mathematical specialties, who study integral calculus as part of a course in mathematical analysis.

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NEW. V.F. Butuzov, N. Ch. Krutitskaya, G.N. Medvedev, A.A. Shishkin. Mathematical analysis in questions and tasks: Textbook. allowance. 5th ed., Rev. 2002 year. 480 pages djvu. 3.8 Mb.
The manual covers all sections of the course of mathematical analysis of functions of one and several variables. For each topic, basic theoretical information is summarized and test questions are suggested; provides solutions to standard and non-standard tasks; tasks and exercises are given for independent work with answers and instructions. Fourth edition 2001
For university students.

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A.A. Burtsev. Methods for solving exam problems in mathematical analysis of the 2nd semester of the 1st year. 2010 year. pdf, 56 pp. 275 Kb.
Variants of tasks for four previous ones. of the year.

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Vinogradova IA et al. Problems and exercises in mathematical analysis (part 1). 1988 year. djvu, 416 pp. 5.0 Mb.
The collection is compiled on the basis of lessons in the course of mathematical analysis in the first year of the Faculty of Mechanics and Mathematics of Moscow State University and reflects the experience of teaching the Department of Mathematical Analysis. It consists of two parts, corresponding to semester I and II. In each part, computational exercises and theoretical problems are separately highlighted. The first part includes the construction of sketches of graphs of functions, calculation of limits, differential calculus of functions of one real variable, theoretical problems. The second part - indefinite integral, definite Riemann integral, differential calculus of functions of several variables, theoretical problems. In chapters containing computational exercises, each paragraph is preceded by detailed methodological instructions. They contain all the definitions used in this section, the formulation of the main theorems, the derivation of some necessary relations, detailed solutions of typical problems are given, and attention is drawn to frequently encountered errors. Most of the problems and exercises are different from the problems contained in the well-known problem book by BP Demidovich. Both parts of the collection include about 1800 exercises for calculations and 350 theoretical problems.

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Vinogradova I. A. et al. Problems and exercises in mathematical analysis (part 2). 1991 year. djvu, 352 p. 3.2 Mb.
The problem book corresponds to the course of mathematical analysis presented in the second year, and contains the following sections: double and triple integrals and their geometric and physical applications, curvilinear and surface integrals of the first and second kind. The necessary theoretical information, typical algorithms suitable for solving entire classes of problems are given, detailed methodological instructions are given.

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Vinogradov et al. Ed. Sadovnichy. Mathematical analysis tasks and exercises. 51 pages PDF. 1.9 Mb.
The section on charting is considered in great detail. 35 pages are occupied by the considered examples.

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Zheltukhin. Indefinite integrals: calculation methods. 2005 year. Size 427 Kb. PDF, 80 pages. A useful guide, can be used as a reference. It not only introduces all the methods for calculating integrals, but also gives a lot of examples for each rule. Recommend.

Download

Zaporozhets. A guide to solving problems in mathematical analysis. 4th ed. 460 pages djvu. 7.7 Mb.
Covers all sections from the study of functions to solving differential equations. A useful book.

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Kalinin, Petrova, Kharin. Indefinite and definite integrals. 2005 year. 230 pages PDF. 1.2 Mb.
Finally, mathematicians began to write books for physicists and other technical students, and not for themselves. I recommend it if you want to learn how to calculate, not prove lemmas and theorems.

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Kalinin, Petrova. Multiple, curvilinear and surface integrals. Tutorial. 2005 year. 230 pages PDF. 1.2 Mb.
This tutorial provides examples of how to calculate various integrals.

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Kaplan. Practical classes in higher mathematics. Analytical geometry, differential calculus, integral calculus, integration of differential equations. In 2 files in one archive. General 925 p. Djvu. 6.9 Mb.
Examples of solving problems throughout the course of general mathematics are considered.

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K.N. Lungu, etc. Collection of problems in higher mathematics. Part 2 for the 2nd course. 2007 year. djvu, 593 pages 4.1 Mb.
Series and integrals. Vector and complex analysis. Differential Equations. Probability theory. Operational calculus. This is not just a book of problems, but also a tutorial. You can use it to learn how to solve problems.

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Lungu, Makarov. Higher mathematics. Guide to solving problems. Part 1. 2005. Size 2.2 Mb. djvu, 315 pp.

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I.A. Maroon. Differential and integral calculus in examples and problems (Functions of one variable). 1970 year. djvu. 400 pages 11.3 MB.
The book is a textbook for solving problems of mathematical analysis (functions of one variable). Contains brief theoretical introductions, solutions to typical examples and tasks for independent solution. In addition to tasks of an algorithmic and computational nature, it contains many tasks that illustrate the theory and contribute to its deeper assimilation, developing independent mathematical thinking of students. The purpose of the book is to teach students to independently solve problems in the course of mathematical analysis

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D.T. Writing. Higher mathematics 100 exam questions. 1999 year. djvu. 304 pp. 9.3 MB.
This manual is intended primarily for students preparing for the exam in higher mathematics in the 1st year. It contains the answers to the exam questions of the oral exam, set out in a concise and accessible form. The manual can be useful for all categories of students studying higher mathematics in one volume or another. It contains the necessary material on 10 sections of the course of higher mathematics, which are usually studied by students in the first year of a university (technical school). Answers to 108 exam questions (with subparagraphs - much more) are accompanied, as a rule, by solving the corresponding examples and problems.

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Sobol B.V., Mishnyakov N.T., Porksheyan V.M. Workshop on higher mathematics. 2006 year. 630 pages djvu. 5.4 Mb.
The book includes all sections of the standard course of higher mathematics for a wide range of specialties in higher educational institutions.
Each chapter (the corresponding section of the course) contains reference material, as well as the basic theoretical provisions necessary for solving problems. A distinctive feature of this publication is a large number of problems with solutions, which allows you to use it not only for classroom studies, but also for students' independent work. Tasks are presented by topic, systematized by methods of solution. Each chapter is completed with a set of self-solving tasks, provided with answers.
The completeness of the presentation of the material and the relative compactness of this publication make it possible to recommend it to teachers and students of higher educational institutions, as well as to students of advanced training institutes who want to systematize their knowledge and skills in this subject.

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E.P. Sulyandziga, G.A. Ushakov. MATH TESTS: LIMIT, DERIVATIVE, ELEMENTS OF ALGEBRA AND GEOMETRY. Uch. allowance. year 2009. pdf, 127 pp. 1.1 Mb.
This tutorial can be thought of as a collection of tasks. The tasks cover traditional topics - the basics of mathematical analysis: a function, its limit and its derivative. There are tasks on the basics of linear algebra and analytical geometry. Since the limit and the derivative of a function are more difficult, and in addition, these topics are fundamental for integral calculus, the greatest attention is paid to them: solutions of typical problems are analyzed in detail. The material collected in the textbook was repeatedly used in practical exercises.
For first year students of all universities.

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• Aleksich G. Problems of convergence of orthogonal series. M .: IL, 1963 (djvu)
• Akhiezer N., Kerin M. On some questions of the theory of moments. Kharkov: GNTIU, 1938 (djvu)
• Akhiezer N.I. The classical problem of moments and some analysis questions related to it. Moscow: Fizmatlit, 1961 (djvu)
• Balk M.B., Petrov V.A., Polukhin A.A. Problem-book on the theory of analytic functions. M .: Education, 1976 (djvu)
• Beckenbach E., Bellman R. Introduction to inequalities. M: World, 1965 (djvu)
• Bernshtein S.N. Extremal properties of polynomials and the best approximation of continuous functions of one real variable. Part 1.L.-M .: GROTL, 1937 (djvu)
• Bermant A.F. The course of mathematical analysis. Part I (12th ed.). M. Fizmatgiz, 1959 (djvu)
• Bermant A.F. The course of mathematical analysis. Part II (9th ed.). M. Fizmatgiz, 1959 (djvu)
• Bermant A.F., Aramanovich I.G. A short course in mathematical analysis for higher educational institutions (5th ed.). Moscow: Nauka, 1967 (djvu)
• M. Brelo, On Topologies and Boundaries in Potential Theory. M .: Mir, 1974 (djvu)
• Brudno A.L. The theory of functions of a real variable. Moscow: Nauka, 1971 (djvu)
• B.M. Budak, S.V. Fomin Multiple integrals and series. Moscow: Nauka, 1965 (djvu)
• A.M. Budylin Fourier series and integrals. L .: SPbGU, 2002 (pdf)
• Bourbaki N. Functions of a real variable. Elementary theory. Moscow: Nauka, 1965 (djvu)
• Baer R. Theory of discontinuous functions. M.-L .: GTTIL, 1932 (djvu)
• Vallee-Poussin C.-J. A course in the analysis of the infinitesimal, volume 1.1922 (djvu)
• Vallee-Poussin C.-J. A course in the analysis of infinitesimal, volume 2.L.-M .: GTTI, 1933 (djvu)
• Vilenkin N.Ya., Bokhan K.A., Maron I.A., Matveev I.V., Smolyansky M.L., Tsvetkov A.T. Problem book on the course of mathematical analysis. Part I. M .: Education, 1971 (djvu)
• Vilenkin N.Ya., Bokhan K.A., Maron I.A., Matveev I.V., Smolyansky M.L., Tsvetkov A.T. Problem book on the course of mathematical analysis. Part II. M .: Education, 1971 (djvu)
• Vulikh B.Z. An Introduction to Functional Analysis (2nd ed.). Moscow: Nauka, 1967 (djvu)
• Vulikh B.Z. A short course in the theory of functions of a real variable. Introduction to Integral Theory (2nd ed.). Moscow: Nauka, 1973 (djvu)
• Vygodsky M. Ya. Handbook of Higher Mathematics (12th ed.). Moscow: Nauka, 1977 (djvu)
• Vygodsky M. Ya. Foundations of Infinitesimal Calculus (3rd ed.). M.-L .: GTTI, 1933 (djvu)
• Hardy G. Integration of elementary functions. M.-L .: ONTI, 1935 (djvu)
• Gelbaum B., Olmsted J. Counterexamples in Analysis. M .: Mir, 1967 (djvu)
• Gelfand I.M., Vilenkin N.Ya. Some applications of harmonic analysis. Equipped Hilbert spaces. (Generic Functions Issue 4). Moscow: Fizmatlit, 1961 (djvu)
• Gelfand I.M., Graev M., Vilenkin N.Ya. Integral geometry and related questions of representation theory. (Generic Functions Issue 5). Moscow: Fizmatlit, 1962 (djvu)
• Gelfand I.M., Graev M., Pyatetskiy-Shapiro I. Representation theory and automorphic functions (Generalized functions, issue 6). Moscow: Fizmatlit, 1966 (djvu)
• Gelfand I.M., Raikov D.A., Shilov G.E. Commutative normed rings. M .: GIFML, 1960 (djvu)
• Gelfand I.M., Shilov G.E. Generalized functions and actions on them (Generalized functions, issue 1) (2nd ed.). Moscow: Fizmatlit, 1959 (djvu)
• Gelfand I.M., Shilov G.E. Core and Generic Function Spaces (Generic Functions Issue 2). Moscow: Fizmatlit, 1958 (djvu)
• Gelfand I.M., Shilov G.E. Some questions of the theory of differential equations (Generalized functions, issue 3). Moscow: Fizmatlit, 1958 (djvu)
• Glivenko V.I. Stieltjes integral. L .: ONTI, 1936 (djvu)
• Gradshtein I. S. Ryzhik I. M. Tables of integrals, sums, series and products (4th ed.). Moscow: Nauka, 1963 (djvu)
• Goursat E. Course of mathematical analysis, volume 1, part 1. Derivatives and differentials. Definite integrals. M.-L .: GTTI, 1933 (djvu)
• Goursat E. Course of mathematical analysis, volume 1, part 2. Series expansions. Geometric applications. M.-L .: GTTI, 1933 (djvu)
• Gursa E. Course of mathematical analysis, volume 2, part 1. Theory of analytic functions. M.-L .: GTTI, 1933 (djvu)
• Gursa E. Course of mathematical analysis, volume 2, part 2. Differential equations. M.-L .: GTTI, 1933 (djvu)
• Gursa E. Course of mathematical analysis, volume 3, part 1. Infinitely close integrals. Partial differential equations. M.-L .: GTTI, 1933 (djvu)
• Gursa E. Course of mathematical analysis, volume 3, part 2. Integral equations. Calculus of variations. M.-L .: GTTI, 1934 (djvu)
• De Bruyne N.G. Asymptotic Methods in Analysis. M .: IL, 1961 (djvu)
• De Ram J. Differentiable manifolds. M .: IL, 1956 (djvu)
• Davydov N.A., Korovkin P.P., Nikolsky V.N. Collection of problems in mathematical analysis (4th ed.). M .: Education, 1973 (djvu)
• Demidovich B.P. (ed.). Tasks and Exercises in Mathematical Analysis for Higher Educational Institutions (6th ed.). Moscow: Nauka, 1968 (djvu)
• Demidovich B.P. (ed.) Problems and exercises in mathematical analysis for high schools (10th ed.). Moscow: Nauka, 1978 (djvu)
• Demidovich B.P. Collection of problems and exercises in mathematical analysis. Moscow: Nauka, 1966 (djvu)
• Demidov A.S. Generalized Functions in Mathematical Physics: Main Ideas and Concepts. New York: Nova Science, 2001 (pdf)
• Jackson D. Fourier series and orthogonal polynomials. M .: IL, 1948 (djvu)
• Jenkins G., Watts D. Spectral analysis and its applications. Issue 1.M .: Mir, 1971 (djvu)
• Jenkins G., Watts D. Spectral analysis and its applications. Issue 2.M .: Mir, 1972 (djvu)
• Dieudonne J. Foundations of modern analysis. M .: Mir, 1964 (djvu)
• Egorova I.A. Practical problem book on mathematical analysis. Part III. Functions of several variables. M .: Uchpedgiz, 1962 (djvu)
• Erugin N.P. Implicit functions. L .: Leningrad State University, 1956 (djvu)
• Zaporozhets G.I. A Guide to Problem Solving in Mathematical Analysis (4th ed.). M .: Higher school, 1966 (djvu)
• Zeldovich B., Myshkis A.D. Elements of Applied Mathematics (3rd ed.). Moscow: Nauka, 1972 (djvu)
• Zeldovich Ya.B., Yaglom I.M. Higher mathematics for beginner physicists and technicians. Moscow: Nauka, 1982 (djvu)
• Sigmund A. Trigonometric series, volume 1. M .: Mir, 1965 (djvu)
• Sigmund A. Trigonometric series, volume 2. M .: Mir, 1965 (djvu)
• Yosida K. Functional Analysis. M .: Mir, 1967 (djvu)
• Kazimirov N.I. Mathematical analysis. Lecture notes for the first year, PetrSU (pdf)
• Kalinin V.V., Petrova I.V., Kharin V.T. Indefinite and definite integrals (Mathematics in oil and gas education, issue 3, part 1). M .: MGUNG im. THEM. Gubkina, 2005 (pdf)
• Kamke E. Lebesgue-Stieltjes Integral. Moscow: Fizmatlit, 1959 (djvu)
• Kaplan I.A. Practical classes in higher mathematics. Parts 1, 2, 3. Analytical geometry on the plane and in space. Differential calculus of functions of one and several independent variables. Integral calculus of functions of one independent variable, integration of differential equations (3rd ed.). Kharkov: KhSU, 1967 (djvu)
• Kaplan I.A. Practical classes in higher mathematics. Part II. Differential calculus of functions of one and several independent variables (5th ed.). Kharkiv: Vishcha school, 1973 (djvu)
• Kaplan I.A. Practical classes in higher mathematics. Part III. Integral calculus of a function of one independent variable. Integration of Differential Equations (4th ed.). Kharkiv: Vishcha school, 1974 (djvu)
• Kaplan I.A. Practical classes in higher mathematics. Part IV. Multiple and Curvilinear Integrals (2nd ed.). Kharkov: KhSU, 1971 (djvu)
• Kaplan I.A. Practical classes in higher mathematics. Part V. Numerical solution of algebraic and transcendental equations, matrix calculus, vector analysis and integration of linear partial differential equations of the first order. (2nd ed.). Kharkiv: KhSU, 1972 (djvu)
• Karlin S., Stadden V. Chebyshev systems and their application in analysis and statistics. Moscow: Nauka, 1976 (djvu)
• Cartan A. Differential calculus. Differential forms. M .: Mir, 1971 (djvu) (djvu)
• Kachenovsky M.I., Bokhan K.M., Karpenko K.M. Collection of tests in mathematical disciplines. Issue I. M .: Uchpedgiz, 1958 (djvu)
• Kozhevnikov N.I., Krasnoshchekova T.I., Shishkin N.E. Fourier series and integral. Field theory. Analytical and special functions. Laplace transform. Moscow: Nauka, 1964 (djvu)
• Collatz L. Functional Analysis and Computational Mathematics. M .: Mir, 1969 (djvu)
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