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E-grāmata: Course in Mathematical Analysis: Volume 2, Metric and Topological Spaces, Functions of a Vector Variable

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(University of Cambridge)
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The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and teachers. Volume 1 focuses on the analysis of real-valued functions of a real variable. This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are developed, with emphasis on their applications to analysis. This leads to the theory of functions of several variables. Differential manifolds in Euclidean space are introduced in a final chapter, which includes an account of Lagrange multipliers and a detailed proof of the divergence theorem. Volume 3 covers complex analysis and the theory of measure and integration.

Papildus informācija

The second volume of three providing a full and detailed account of undergraduate mathematical analysis.
Introduction ix
Part Three Metric and topological spaces 301(182)
11 Metric spaces and normed spaces
303(27)
11.1 Metric spaces: examples
303(6)
11.2 Normed spaces
309(3)
11.3 Inner-product spaces
312(5)
11.4 Euclidean and unitary spaces
317(2)
11.5 Isometries
319(4)
11.6 *The Mazur-Ulam theorem*
323(4)
11.7 The orthogonal group Od
327(3)
12 Convergence, continuity and topology
330(23)
12.1 Convergence of sequences in a metric space
330(7)
12.2 Convergence and continuity of mappings
337(5)
12.3 The topology of a metric space
342(7)
12.4 Topological properties of metric spaces
349(4)
13 Topological spaces
353(33)
13.1 Topological spaces
353(8)
13.2 The product topology
361(5)
13.3 Product metrics
366(4)
13.4 Separation properties
370(5)
13.5 Countability properties
375(4)
13.6 *Examples and counterexamples*
379(7)
14 Completeness
386(45)
14.1 Completeness
386(9)
14.2 Banach spaces
395(5)
14.3 Linear operators
400(6)
14.4 *Tietze's extension theorem*
406(2)
14.5 The completion of metric and normed spaces
408(4)
14.6 The contraction mapping theorem
412(8)
14.7 *Baire's category theorem*
420(11)
15 Compactness
431(33)
15.1 Compact topological spaces
431(4)
15.2 Sequentially compact topological spaces
435(4)
15.3 Totally bounded metric spaces
439(2)
15.4 Compact metric spaces
441(4)
15.5 Compact subsets of C(K)
445(3)
15.6 *The Hausdorff metric*
448(4)
15.7 Locally compact topological spaces
452(5)
15.8 Local uniform convergence
457(3)
15.9 Finite-dimensional normed spaces
460(4)
16 Connectedness
464(19)
16.1 Connectedness
464(6)
16.2 Paths and tracks
470(3)
16.3 Path-connectedness
473(2)
16.4 *Hilbert's path*
475(3)
16.5 *More space-filling paths*
478(2)
16.6 Rectifiable paths
480(3)
Part Four Functions of a vector variable 483(108)
17 Differentiating functions of a vector variable
485(28)
17.1 Differentiating functions of a vector variable
485(6)
17.2 The mean-value inequality
491(5)
17.3 Partial and directional derivatives
496(4)
17.4 The inverse mapping theorem
500(2)
17.5 The implicit function theorem
502(2)
17.6 Higher derivatives
504(9)
18 Integrating functions of several variables
513(32)
18.1 Elementary vector-valued integrals
513(2)
18.2 Integrating functions of several variables
515(2)
18.3 Integrating vector-valued functions
517(8)
18.4 Repeated integration
525(5)
18.5 Jordan content
530(4)
18.6 Linear change of variables
534(2)
18.7 Integrating functions on Euclidean space
536(1)
18.8 Change of variables
537(6)
18.9 Differentiation under the integral sign
543(2)
19 Differential manifolds in Euclidean space
545(46)
19.1 Differential manifolds in Euclidean space
545(3)
19.2 Tangent vectors
548(4)
19.3 One-dimensional differential manifolds
552(3)
19.4 Lagrange multipliers
555(10)
19.5 Smooth partitions of unity
565(3)
19.6 Integration over hypersurfaces
568(4)
19.7 The divergence theorem
572(10)
19.8 Harmonic functions
582(5)
19.9 Curl
587(4)
Appendix B Linear algebra 591(10)
B.1 Finite-dimensional vector spaces
591(3)
B.2 Linear mappings and matrices
594(3)
B.3 Determinants
597(2)
B.4 Cramer's rule
599(1)
B.5 The trace
600(1)
Appendix C Exterior algebras and the cross product 601(6)
C.1 Exterior algebras
601(3)
C.2 The cross product
604(3)
Appendix D Tychonoff's theorem 607(5)
Index 612(6)
Contents for Volume I 618(3)
Contents for Volume III 621
D. J. H. Garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge. He has 50 years' experience of teaching undergraduate students in most areas of pure mathematics, but particularly in analysis.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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