In this post we will see the second part of Course in Mathematical
Analysis by S. M. Nikolsky.

The major part of this two-volume textbook stems from the
course in mathematical analysis given by the author for many
years at the Moscow Physico-technical Institute.

The first volume consisting of eleven chapters includes an
introduction (Chapter 1)which treats offundamental notions of
mathematical analysis using an intuitive concept ofa limit. With
the aid of visual interpretation and some considerations of a
physical character it establishes the relationship between the
derivative and the integral and gives some elements of differentiation
and integration techniques necessary to those readers
who are simultaneously studying physics.

The notion of a real number is interpreted in the first volume
(Chapter 2) on the basis of its representation as an infinite decimal.
Chapters 3-11 contain the following topics: Limit of Sequence,
Limit of Function, Functions of One Variable, Functions
of Several Variables, Indefinite Integral, Definite Integral,
Some Applications of Integrals, Series.

This book was translated from the Russian by V. M. Volosov. The
book was published by first Mir Publishers in 1977 with reprints in
1981, 1985 and 1987. The copy below is from the 1987 print.

All credits to the original uploader.

Update 18 December 2021: New post with new links to both volumes here.

Table of Contents

Chapter 12. Multiple Integrals 9

§ 12.1. Introduction 9
§ 12.2. Jordan Squarable Sets 11
§ 12.3. Some Important Examples of Squarable Sets 17
§ 12.4. One More Test for Measurability of a Set. Area in Polar Coordinates. 19
§ 12.5. Jordan Measurable Three-dimensional and n-dimensional Sets. 20
§ 12.6. The Notion of Multiple Integral  24
§ 12.7. Upper and Lower Integral Sums. Key Theorem 27
§ 12.8. Integrability of a Continuous Function on a Measurable Closed Set.
Some Other Integrability Conditions    32
§ 12.9. Set of Lebesgue Measure Zero  34
§ 12.10. Proof ofLebesgue’s Theorem. Connection Between Integrability and
Boundedness of a Function 35
§ 12.11. Properties of Multiple Integrals 38
§ 12.12. Reduction of Multiple Integral to Iterated Integral 41
§ 12.13. Continuity of Integral Dependent on Parameter 48
§ 12.14. Geometrical Interpretation of the Sign of a Determinant 51
§ 12.15. Change of Variables in Multiple Integral. Simplest Case  54
§ 12.16. Change of Variables in Multiple Integral. General Case  56
§ 12.17. Proof of Lemma 1, § 12.16 59
§ 12.18. Double Integral in Polar Coordinates. 63
§ 12.19. Triple Integral in Spherical Coordinates 65
§ 12.20. General Properties of Continuous Operators 67
§ 12.21. More on Change of Variables in Multiple Integral        68
§ 12.22. Improper Integral with Singularities on the Boundary of the Domain
of Integration. Change of Variables 71
§ 12.23. Surface Area  73

Chapter 13. Scalar and Vector Fields. Differentiation and Integration
of Integral
with Respect to Parameter. Improper Integrals     80

§ 13.1. Line Integral of the First Type  80
§ 13.2. Line Integral of the Second Type            81
§ 13.3. Potential of a Vector Field  83
§ 13.4. Orientation of a Domain in the Plane 91
§ 13.5. Green’s Formula. Computing Area with the Aid of Line Integral  92
§ 13.6. Surface Integral of the First Type  96
§ 13.7. Orientation of a Surface  98
§ 13.8. Integral over an Oriented Domain in the Plane          102
§ 13.9. Flux of a Vector Through an Oriented Surface 104
§ 13.10. Divergence. Gauss-Ostrogradsky Theorem 107
§ 13.11. Rotation of a Vector. Stokes’ Theorem. 114
§ 13.12. Differentiation of Integral with Respect to Parameter 118
§ 13.13. Improper Integrals 121
§ 13.14. Uniform Convergence of Improper Integrals 128
§ 13.15. Uniformly Convergent Integral over Unbounded Domain. 135
§ 13.16. Uniformly Convergent Improper Integral with Variable Singularity 140

Chapter 14. Normed Linear Spaces. Orthogonal Systems 147

§ 14.1. Space C of Continuous Functions. 147
§ 14.2. Spaces L’, L’_p and l_p 149
§ 14.3. Spaces L_2 and L’_2  154
§ 14.4. Approximation with Finite Functions      156
§ 14.:5. Linear Spaces. Fundamentals ofthe Theory ofNormed Linear Spaces 163
§ 14.6. Orthogonal Systems in Space with Scalar Product 170
§ 14.7. Orthogonalization Process            181
§ 14.8. Properties of Spaces L’_2(\Omega) and L_2(\Omega) . 185
§ 14.9. Complete Systems of Functions in the Spaces C, L’_2 and L’ (L_2, L) 187

Chapter 15. Fourier Series. Approximation of Functions with Polynomials   188

§ 15.1. Preliminaries   188
§ I5.2. Dirichlet’s Sum 195
§ 15.3. Formulas for the Remainder of Fourier’s Series 197
§ 15.4. Oscillation Lemmas  199
§ 15.5. Test for Convergence of Fourier Series. Completeness of Trigonometric
System of Functions 203
§ 15.6. Complex Form of Fourier Series 211
§ 15.7. Differentiation and Integration of Fourier Series  213
§ 15.8. Estimating the Remainder of Fourier’s Series 216
§ 15.9. Gibbs’ Phenomenon                   217
§ 15.10. Fejer’g Sums               221
§ 15.11. Elements of the Theory of Fourier Series for Functions of Several
Variables. 225
§ 15.12. Algebraic Polynomials. Chebyshev’s Polynomials     235
§ 15.13. Weierstrass’ Theorem   236
§ 15.14. Legendre’s Polynomials 237

Chapter 16. Fourier Integral. Generalized Functions 240
§ 16.1. Notion of Fourier Integral 240
§ 16.2. Lemma on Change of Order of Integration 243
§ 16.3. Convergence of Fourier’s Single Integral 245
§ 16.4. Fourier Transform and Its Inverse. Iterated Fourier
Integral. Fourier Cosine and Sine Transforms 247
§ 16.5. Differentiation and Fourier Transformation It 249
§ 16.6. Space S 250
§ 16.7. Space S’ of Generalized Functions 255
§ 16.8. Many-dimensional Fourier Integrals and Generalized Functions  265
§ 16.9. Finite Step Functions. Approximation in the Mean Square 273
§ 16.10. Plancherel’s Theorem. Estimating Speed of Convergence of Fourier’s
Integrals 278
§ 16.11. Generalized Periodic Functions 283

Chapter 17. Differentiable Manifolds and Differential Forms 289
§ 17.1. Differentiable Manifolds 289
§ 17.2. Boundary of a Differentiable Manifold and Its Orientation 299
§ 17.3. Differential Forms. 310
§ 17.4. Stokes’ Theorem 220

Chapter 18. Supplementary Topics 326
§ 18.1. Generalized Minkowski’s Inequality 326
§ 18.2. Sobolev’s Regularization of Function 329
§ 18.3. Convolution 333
§ 18.4. Partition of Unity 335

Chapter 19. Lebesgue Integral  338

§ 19.1. Lebesgue Mea.sure  338
§ 19.2. Measurable Functions  348
§ 19.3. Lebesgue lntegral 35S
§ 19.4. Lebesgue Integral on Unbounded Set 388
§ 19.5. Sobolev’s Generalized Derivative  392
§ 19.6. Space D’ of Generalized Functions 404
§ 19.7. Incompleteness of Space L 407
§ 19.8. Generalization of Jordan Measure 408
§ 19.9. Riemann-Stieltjes Integral  414
§ 19.10. Stieltjes Integral  415
§ 19.11. Generalization of Lebesgue Integral 423
§ 19.12 Lebesgue-StieJtjes Integral 424
§ 19.13. Extension of Functions. Weierstrass’ Theorem 433
Name Index 437
Subject Index  438