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**

Â§ 12.1. Introduction 9

Â§ 12.2. Jordan Squarable Sets 11

Â§ 12.3. Some Important Examples of Squarable Sets 17

Â§ 12.4. One More Test for Measurability of a Set. Area in Polar Coordinates. 19

Â§ 12.5. Jordan Measurable Three-dimensional and n-dimensional Sets. 20

Â§ 12.6. The Notion of Multiple Integral 24

Â§ 12.7. Upper and Lower Integral Sums. Key Theorem 27

Â§ 12.8. Integrability of a Continuous Function on a Measurable Closed Set.

Some Other Integrability Conditions 32

Â§ 12.9. Set of Lebesgue Measure Zero 34

Â§ 12.10. Proof ofLebesgue’s Theorem. Connection Between Integrability and

Boundedness of a Function 35

Â§ 12.11. Properties of Multiple Integrals 38

Â§ 12.12. Reduction of Multiple Integral to Iterated Integral 41

Â§ 12.13. Continuity of Integral Dependent on Parameter 48

Â§ 12.14. Geometrical Interpretation of the Sign of a Determinant 51

Â§ 12.15. Change of Variables in Multiple Integral. Simplest Case 54

Â§ 12.16. Change of Variables in Multiple Integral. General Case 56

Â§ 12.17. Proof of Lemma 1, Â§ 12.16 59

Â§ 12.18. Double Integral in Polar Coordinates. 63

Â§ 12.19. Triple Integral in Spherical Coordinates 65

Â§ 12.20. General Properties of Continuous Operators 67

Â§ 12.21. More on Change of Variables in Multiple Integral 68

Â§ 12.22. Improper Integral with Singularities on the Boundary of the Domain

of Integration. Change of Variables 71

Â§ 12.23. Surface Area 73

**Chapter 13. Scalar and Vector Fields. Differentiation and Integration**

** of Integral**

** with Respect to Parameter. Improper Integrals 80**

Â§ 13.1. Line Integral of the First Type 80

Â§ 13.2. Line Integral of the Second Type 81

Â§ 13.3. Potential of a Vector Field 83

Â§ 13.4. Orientation of a Domain in the Plane 91

Â§ 13.5. Green’s Formula. Computing Area with the Aid of Line Integral 92

Â§ 13.6. Surface Integral of the First Type 96

Â§ 13.7. Orientation of a Surface 98

Â§ 13.8. Integral over an Oriented Domain in the Plane 102

Â§ 13.9. Flux of a Vector Through an Oriented Surface 104

Â§ 13.10. Divergence. Gauss-Ostrogradsky Theorem 107

Â§ 13.11. Rotation of a Vector. Stokes’ Theorem. 114

Â§ 13.12. Differentiation of Integral with Respect to Parameter 118

Â§ 13.13. Improper Integrals 121

Â§ 13.14. Uniform Convergence of Improper Integrals 128

Â§ 13.15. Uniformly Convergent Integral over Unbounded Domain. 135

Â§ 13.16. Uniformly Convergent Improper Integral with Variable Singularity 140

**Chapter 14. Normed Linear Spaces. Orthogonal Systems 147**

Â§ 14.1. Space C of Continuous Functions. 147

Â§ 14.2. Spaces L’, L’_p and l_p 149

Â§ 14.3. Spaces L_2 and L’_2 154

Â§ 14.4. Approximation with Finite Functions 156

Â§ 14.:5. Linear Spaces. Fundamentals ofthe Theory ofNormed Linear Spaces 163

Â§ 14.6. Orthogonal Systems in Space with Scalar Product 170

Â§ 14.7. Orthogonalization Process 181

Â§ 14.8. Properties of Spaces L’_2(\Omega) and L_2(\Omega) . 185

Â§ 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**

Â§ 15.1. Preliminaries 188

Â§ I5.2. Dirichlet’s Sum 195

Â§ 15.3. Formulas for the Remainder of Fourier’s Series 197

Â§ 15.4. Oscillation Lemmas 199

Â§ 15.5. Test for Convergence of Fourier Series. Completeness of Trigonometric

System of Functions 203

Â§ 15.6. Complex Form of Fourier Series 211

Â§ 15.7. Differentiation and Integration of Fourier Series 213

Â§ 15.8. Estimating the Remainder of Fourier’s Series 216

Â§ 15.9. Gibbs’ Phenomenon 217

Â§ 15.10. Fejer’g Sums 221

Â§ 15.11. Elements of the Theory of Fourier Series for Functions of Several

Variables. 225

Â§ 15.12. Algebraic Polynomials. Chebyshev’s Polynomials 235

Â§ 15.13. Weierstrass’ Theorem 236

Â§ 15.14. Legendre’s Polynomials 237

**Chapter 16. Fourier Integral. Generalized Functions 240**

Â§ 16.1. Notion of Fourier Integral 240

Â§ 16.2. Lemma on Change of Order of Integration 243

Â§ 16.3. Convergence of Fourier’s Single Integral 245

Â§ 16.4. Fourier Transform and Its Inverse. Iterated Fourier

Integral. Fourier Cosine and Sine Transforms 247

Â§ 16.5. Differentiation and Fourier Transformation It 249

Â§ 16.6. Space S 250

Â§ 16.7. Space S’ of Generalized Functions 255

Â§ 16.8. Many-dimensional Fourier Integrals and Generalized Functions 265

Â§ 16.9. Finite Step Functions. Approximation in the Mean Square 273

Â§ 16.10. Plancherel’s Theorem. Estimating Speed of Convergence of Fourier’s

Integrals 278

Â§ 16.11. Generalized Periodic Functions 283

**Chapter 17. Differentiable Manifolds and Differential Forms 289**

Â§ 17.1. Differentiable Manifolds 289

Â§ 17.2. Boundary of a Differentiable Manifold and Its Orientation 299

Â§ 17.3. Differential Forms. 310

Â§ 17.4. Stokes’ Theorem 220

**Chapter 18. Supplementary Topics 326**

Â§ 18.1. Generalized Minkowski’s Inequality 326

Â§ 18.2. Sobolev’s Regularization of Function 329

Â§ 18.3. Convolution 333

Â§ 18.4. Partition of Unity 335

**Chapter 19. Lebesgue Integral 338**

Â§ 19.1. Lebesgue Mea.sure 338

Â§ 19.2. Measurable Functions 348

Â§ 19.3. Lebesgue lntegral 35S

Â§ 19.4. Lebesgue Integral on Unbounded Set 388

Â§ 19.5. Sobolev’s Generalized Derivative 392

Â§ 19.6. Space D’ of Generalized Functions 404

Â§ 19.7. Incompleteness of Space L 407

Â§ 19.8. Generalization of Jordan Measure 408

Â§ 19.9. Riemann-Stieltjes Integral 414

Â§ 19.10. Stieltjes Integral 415

Â§ 19.11. Generalization of Lebesgue Integral 423

Â§ 19.12 Lebesgue-StieJtjes Integral 424

Â§ 19.13. Extension of Functions. Weierstrass’ Theorem 433

Name Index 437

Subject Index 438

great post, thanks a lot, waiting for vol. 1???????

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Vol. 2 we have found on the net. We both Vol. 1 and 2 in hard copy. But Vol. 1 is not scanned yet by us. If any one finds links to Vol. 1 it would be great.

D

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Damitr where can i get the hard copy of this book’s volume 1. Please let me know if you are aware of it. Thanks…

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Hi, I am unable to download any of the latest books that have been posted here.

are the links unavailable now?

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Thanx for the upload! Hope you would upload the Volume 1 soon.

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When I click “request download file” it says “no such file”. Could you reupload the file please? Thank you.

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Thanks Id for pointing this.I will soon post the 4shared links and will update.

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id , I have re-uploaded the missing files in 4shared ,please check and let me know if the problem exists

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Hi desperadomar, thank you very much for reuploading the file. I can download it now.

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Nikolsky’s this book is great. But, in all the odd pages of this vol2 ebook edition, the words on the left are somewhat out of shape. Could someone re-scan the vol2 ?

This is a great book. I read the vol1 a few years ago. Some theorems about integral in vol1 are the exercises of Zorich’s Mathematical analysis. And Nikolsky’s this book is the reference book of Department of Mathematics of Moscow State University.

The reference books of mathematical analysis at Department of Mathematics of Moscow Sate University are:

1 V.A.Zorich, mathematical analysis

2 J.Dieudonne, Elements de Analyse, Gauthier-Villars, 1969

3 Valle Possin, Cours de Analyse Infinitesimale, Gauthier-Villars, 1903

4 S.M.Nikolsky, a course in mathematical analysis

5 L.D.Kudryavsev, a course of mathematical analysis(in Russian)

6 A.N.Kolmogorov, P.S.Aleksandrov, introduction functions of real variables(in Russian)

7 L.Schwartz, Cours de Analyse, Hermann, 1981

8 B.P.Demidovich, problems in mathematical analysis

9 L.D.Kudryavtsev, problems in mathematical analysis(in Russian)

L.Schwartz, Cours d’analyse (vo1 and vol2) is also a very good book. It has ebook edition, but vol1(I remember it is vol1, if not, it is vol2) is incorrectly ocr into vector pdf. Hope someone who like this book can re-scan the incorrectly ocred vol.

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Thanks for the list of books at your department.

The Vol. 2 was picked up from the net so that is what was there.

But soon we may have both the volumes scanned again, in some time, we have them in hard copy.

Also

8 B.P.Demidovich, problems in mathematical analysisis already posted here.We might have soon books in the Topics in Mathematics, and PLM posted.

Also a good list and comprehensive of the

Little Russian Books on Mathematicscan be found here, we have collected many of these and they will be soon posted.Any help regarding posting of books that you have listed is appreciated.

D

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Hey I have book by prasolov on problems in plane geometry which have about 3000-6000 problems solely on plane geometry .Due to low Internet connection I can’t upload but I can email it to you if possible.

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Devarsh may you email it to us too!

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Dear Devarsh,

please email the same.

thanks in advance.

shrikant

djhala2016@yahoo.com

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i am waiting for volume 1. Please upload it soon. Thank You.

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I know this is not very useful, but I happen to possess the Russian version of the first volume of this book.

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may you write for us, the table of contents? please.

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Hey Carlos i saw the first volume of this book in my local library. If you want pics of the contents provide your email ID..

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can you upload the russian version so at least the problems given in the book can be useful

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Pls go here http://www.new.dli.ernet.in/scripts/FullindexDefault.htm?path1=/data14/upload/0018/334&first=1&last=434&barcode=99999990327504 to get the volume one of this book. I have volume 2 of this book but searching for volume 1 in hard copy.

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I can’t access this book. 😦

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