S. M. NIKOLSKY A C o u r s e o f M a t h e m a t i c a l A n a l y s i s Volume 2 M I R P U B L I S H E R S M O S C O W Academician S.M NIKOLSKY a State Prize winner, the author of more than 130 scienti�c papers and several monographs including Quadrature Formulas, Approximation of Functions of Several Variables and Embedding Theorems and Integral Representation of Functions and Embedding Theorems (co-author). He contributed many fundamental results to the theory of approximation of functions, variational methods for solving boundary-value problems and functional analysis. For his monograph Approximation of Functions of Several Variables and Embedding Theorems he was awarded the Chebyshev Prize of the USSR Academy of Sciences. A Course of Mathematical Analysis C. M. HHKOJIbCKHft KYPC M ATEMATMM ECKOrO AHAJ1H3A TOM 2 H3flATEJILCrBO «HAYKA» MOCKBA S . M . N IK O L SK Y Member, USSR Academy of Sciences A Course o f Mathematical Analysis i V olume 2 Translated from the Russian by V. M. VOLOSOV, D. Sc. MIR PUBLISHERS MOSCOW First published 1977 Second printing 1981 Third printing 1985 Fourth printing 1987 TO THE READER Mir Publishers would be grateful for your comments on the content, translation and design of this book. We would also be pleased to receive any other sugges­ tions you may wish to make. Our address is: USSR, 129820, Moscow 1-110, GSP Pervy Rizhsky Pereulok, 2 MIR PUBLISHERS Ha QH2AUUCKOM H3HK€ © H w ienbciBo «HayKa», 1975 © English translation, Mir Publishers, 1977 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 /i-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 of Lebesgue’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 C ase................. 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 5