Verfügbarkeit: Postmodern analysis | Technische Universität München

Postmodern analysis:

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Bibliographische Detailangaben
Beteilige Person:	Jost, Jürgen 1956- (VerfasserIn)
Format:	Buch
Sprache:	Englisch Deutsch
Veröffentlicht:	Berlin [u.a.] Springer 2003
Ausgabe:	2. ed.
Schriftenreihe:	Universitext
Schlagwörter:	Analisi matematica Analysis Mathematical analysis
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009852736&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XVII, 367 S.
ISBN:	3540438734

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Datensatz im Suchindex

DE-BY-TUM_call_number	0001 08.2003 A 84
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adam_text	JUERGEN JOST POSTMODERN ANALYSIS SECOND EDITION SPRINGER CONTENTS CHAPTER I. CALCULUS FOR FUNCTIONS OF ONE VARIABLE 0. PREREQUISITES PROPERTIES OF THE REAL NUMBERS, LIMITS AND CONVERGENCE OF SEQUENCES OF REAL NUMBERS, EXPONENTIAL FUNCTION AND LOGARITHM. EXERCISES 1. ......................................................... 3 LIMITS AND CONTINUITY OF FUNCTIONS DEFINITIONS OF CONTINUITY, UNIFORM CONTINUITY, PROPERTIES OF CONTINUOUS FUNCTIONS, INTERMEDIATE VALUE THEOREM, HOLDER AND LIPSCHITZ CONTINUITY. EXERCISES 2. DIFFERENTIABILITY ......................................................... 13 DEFINITIONS OF DIFFERENTIABILITY, DIFFERENTIATION RULES, DIFFERENTIABLE FUNCTIONS ARE CONTINUOUS, HIGHER ORDER DERIVATIVES. EXERCISES 3. ......................................................... 21 CHARACTERISTIC PROPERTIES OF DIFFERENTIABLE FUNCTIONS. DIFFERENTIAL EQUATIONS CHARACTERIZATION OF LOCAL EXTREMA BY THE VANISHING OF THE DERIVATIVE, MEAN VALUE THEOREMS, THE DIFFERENTIAL EQUATION F* UNIQUENESS OF SOLUTIONS OF DIFFERENTIAL EQUATIONS, QUALITATIVE BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS AND INEQUALITIES, CHARACTERIZATION OF LOCAL MAXIMA AND MINIMA VIA SECOND DERIVATIVES, TAYLOR EXPANSION. EXERCISES 4. THE BANACH FIXED POINT THEOREM. THE CONCEPT OF BANACH SPACE ......................................................... 31 BANACH FIXED POINT THEOREM, DEFINITION OF NORM, METRIC, CAUCHY SEQUENCE, COMPLETENESS. EXERCISES ......................................................... 43 XIV CONTENTS 5. UNIFORM CONVERGENCE. INTERCHANGEABILITY OF LIMITING PROCESSES. EXAMPLES OF BANACH SPACES. THE THEOREM OF ARZELA-ASCOLI CONVERGENCE OF SEQUENCES OF FUNCTIONS, POWER SERIES, CONVERGENCE THEOREMS, UNIFORMLY CONVERGENT SEQUENCES, NORMS ON FUNCTION SPACES, THEOREM OF ARZELA-ASCOLI ON THE UNIFORM CONVERGENCE OF SEQUENCES OF UNIFORMLY BOUNDED AND EQUICONTINUOUS FUNCTIONS. EXERCISES 6. ......................................................... 47 INTEGRALS AND ORDINARY DIFFERENTIAL EQUATIONS PRIMITIVES, RIEMANN INTEGRAL, INTEGRATION RULES, INTEGRATION BY PARTS, CHAIN RULE, MEAN VALUE THEOREM, INTEGRAL AND AREA, ODES, THEOREM OF PICARD-LINDELOEF ON THE LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS OF ODES WITH A LIPSCHITZ CONDITION. EXERCISES ......................................................... 61 CHAPTER 11. TOPOLOGICAL CONCEPTS 7. METRIC SPACES: CONTINUITY, TOPOLOGICAL NOTIONS, COMPACT SETS DEFINITION OF A METRIC SPACE, OPEN, CLOSED, CONVEX, CONNECTED, COMPACT SETS, SEQUENTIAL COMPACTNESS, CONTINUOUS MAPPINGS BETWEEN METRIC SPACES, BOUNDED LINEAR OPERATORS, EQUIVALENCE OF NORMS IN RD, DEFINITION OF A TOPOLOGICAL SPACE. EXERCISES ......................................................... 77 CHAPTER III. CALCULUS IN EUCLIDEAN AND BANACH SPACES 8. DIFFERENTIATION IN BANACH SPACES DEFINITION OF DIFFERENTIABILITY OF MAPPINGS BETWEEN BANACH SPACES, DIFFERENTIATION RULES, HIGHER DERIVATIVES, TAYLOR EXPANSION. EXERCISES ......................................................... 103 9. DIFFERENTIAL CALCULUS IN RD A. SCALAR VALUED FUNCTIONS GRADIENT, PARTIAL DERIVATIVES, HESSIAN, LOCAL EXTREMA, LAPLACE B. VECTOR VALUED FUNCTIONS JACOBI MATRIX, VECTOR FIELDS, DIVERGENCE, ROTATION. EXERCISES OPERATOR, PARTIAL DIFFERENTIAL EQUATIONS ......................................................... 115 10. THE IMPLICIT FUNCTION THEOREM. APPLICATIONS IMPLICIT AND INVERSE FUNCTION THEOREMS, EXTREMA WITH CONSTRAINTS, LAGRANGE MULTIPLIERS. EXERCISES ......................................................... 133 CONTENTS XV 11. CURVES IN RD. SYSTEMS OF ODES REGULAR AND SINGULAR CURVES, LENGTH, RECTIFIABILITY, ARCS, JORDAN ARC THEOREM, HIGHER ORDER ODE AS SYSTEMS OF ODES. EXERCISES ......................................................... 145 CHAPTER IV. THE LEBESGUE INTEGRAL 12. PREPARATIONS. SEMICONTINUOUS FUNCTIONS THEOREM OF DINI, UPPER AND LOWER SEMICONTINUOUS FUNCTIONS, THE CHARACTERISTIC FUNCTION OF A SET. EXERCISES 13. THE LEBESGUE INTEGRAL FOR SEMICONTINUOUS FUNCTIONS. ......................................................... 157 THE VOLUME OF COMPACT SETS THE INTEGRAL OF CONTINUOUS AND SEMICONTINUOUS FUNCTIONS, THEOREM OF FUBINI, VOLUME, INTEGRALS OF ROTATIONALLY SYMMETRIC FUNCTIONS AND OTHER EXAMPLES. EXERCISES 14. LEBESGUE INTEGRABLE FUNCTIONS AND SETS ......................................................... 165 UPPER AND LOWER INTEGRAL, LEBESGUE INTEGRAL, APPROXIMATION OF LEBESGUE INTEGRALS, INTEGRABILITY OF SETS. EXERCISES 15. NULL FUNCTIONS AND NULL SETS. THE THEOREM OF FUBINI ......................................................... 183 NULL FUNCTIONS, NULL SETS, CANTOR SET, EQUIVALENCE CLASSES OF INTEGRABLE FUNCTIONS, THE SPACE L1 , FUBINIS THEOREM FOR INTEGRABLE FUNCTIONS. EXERCISES 16. THE CONVERGENCE THEOREMS OF LEBESGUE INTEGRATION ......................................................... 195 THEORY MONOTONE CONVERGENCE THEOREM OF B. LEVI, FATOUS LEMMA, DOMINATED CONVERGENCE THEOREM OF H. LEBESGUE, PARAMETER DEPENDENT INTEGRALS, DIFFERENTIATION UNDER THE INTEGRAL SIGN. EXERCISES 17. MEASURABLE FUNCTIONS AND SETS. JENSENS INEQUALITY. ......................................................... 205 THE THEOREM OF EGOROV MEASURABLE FUNCTIONS AND THEIR PROPERTIES, MEASURABLE SETS, MEASURABLE FUNCTIONS AS LIMITS OF SIMPLE FUNCTIONS, THE COMPOSITION OF A MEASURABLE FUNCTION WITH A CONTINUOUS FUNCTION IS MEASURABLE, JENSENS INEQUALITY FOR CONVEX FUNCTIONS, THEOREM OF EGOROV ON ALMOST UNIFORM CONVERGENCE OF MEASURABLE FUNCTIONS, THE ABSTRACT CONCEPT OF A MEASURE. EXERCISES ......................................................... 217 XVI CONTENTS 18. THE TRANSFORMATION FORMULA TRANSFORMATION OF MULTIPLE INTEGRALS UNDER DIFFEOMORPHISMS, INTEGRALS IN POLAR COORDINATES. EXERCISES ......................................................... 229 CHAPTER V. LP AND SOBOLEV SPACES 19. THE LP-SPACES LP-FUNCTIONS, HOLDERS INEQUALITY, MINKOWSKIS INEQUALITY, COMPLETENESS OF LP-SPACES, CONVOLUTIONS WITH LOCAL KERNELS, LEBESGUE POINTS, APPROXIMATION OF LP-FUNCTIONS BY SMOOTH FUNCTIONS THROUGH MOLLIFICATION, TEST FUNCTIONS, PARTITIONS OF UNITY. EXERCISES 20. INTEGRATION BY PARTS. WEAK DERIVATIVES. SOBOLEV SPACES WEAK DERIVATIVES DEFINED BY AN INTEGRATION BY PARTS FORMULA, SOBOLEV ......................................................... 241 FUNCTIONS HAVE WEAK DERIVATIVES IN LP-SPACES, CALCULUS FOR SOBOLEV FUNCTIONS, SOBOLEV EMBEDDING THEOREM ON THE CONTINUITY OF SOBOLEV FUNCTIONS WHOSE WEAK DERIVATIVES ARE INTEGRABLE TO A SUFFICIENTLY HIGH POWER, POINCARE INEQUALITY, COMPACTNESS THEOREM OF RELLICH-KONDRACHOV ON THE LP-CONVERGENCE OF SEQUENCES WITH BOUNDED SOBOLEV NORM. EXERCISES ......................................................... 261 CHAPTER VI. INTRODUCTION TO THE CALCULUS OF VARIATIONS AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 21. HILBERT SPACES. WEAK CONVERGENCE THEOREM, WEAK CONVERGENCE, WEAK COMPACTNESS OF BOUNDED SEQUENCES, BANACH-SAKS LEMMA ON THE CONVERGENCE OF CONVEX COMBINATIONS OF BOUNDED SEQUENCES. EXERCISES 22. VARIATIONAL PRINCIPLES AND PARTIAL DIFFERENTIAL EQUATIONS DEFINITION AND PROPERTIES OF HILBERT SPACES, RIESZ REPRESENTATION ......................................................... 285 DIRICHLETS PRINCIPLE, WEAKLY HARMONIC FUNCTIONS, DIRICHLET PROBLEM, EULER-LAGRANGE EQUATIONS, VARIATIONAL PROBLEMS, WEAK LOWER SEMICONTINUITY OF VARIATIONAL INTEGRALS WITH CONVEX INTEGRANDS, EXAMPLES FROM PHYSICS AND CONTINUUM MECHANICS, HAMILTONS PRINCIPLE, EQUILIBRIUM STATES, STABILITY, THE LAPLACE OPERATOR IN POLAR COORDINATES. EXERCISES 23. REGULARITY OF WEAK SOLUTIONS ......................................................... 295 SMOOTHNESS OF WEAKLY HARMONIC FUNCTIONS AND OF WEAK SOLUTIONS OF GENERAL ELLIPTIC PDES, BOUNDARY REGULARITY, CLASSICAL SOLUTIONS. EXERCISES ......................................................... 327 CONTENTS XVII 24. THE MAXIMUM PRINCIPLE WEAK AND STRONG MAXIMUM PRINCIPLE FOR SOLUTIONS OF ELLIPTIC PDES, BOUNDARY POINT LEMMA OF E. HOPF, GRADIENT ESTIMATES, THEOREM OF LIOUVILLE. EXERCISES 25. THE EIGENVALUE PROBLEM FOR THE LAPLACE OPERATOR BASIS OF L2 AS AN APPLICATION OF THE RELLICH COMPACTNESS THEOREM. EXERCISES ......................................................... 355 INDEX OF NOTATION .......................................... 361 INDEX ....................................................... 363 ......................................................... 343 EIGENFUNCTIONS OF THE LAPLACE OPERATOR FORM A COMPLETE ORTHONORMAL
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author	Jost, Jürgen 1956-
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discipline	Mathematik
edition	2. ed.
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illustrated	Not Illustrated
indexdate	2024-12-20T11:03:52Z
institution	BVB
isbn	3540438734
language	English German
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spellingShingle	Jost, Jürgen 1956- Postmodern analysis Analisi matematica sbt Analysis Mathematical analysis Analysis (DE-588)4001865-9 gnd
subject_GND	(DE-588)4001865-9
title	Postmodern analysis
title_alt	Postmoderne Analysis
title_auth	Postmodern analysis
title_exact_search	Postmodern analysis
title_full	Postmodern analysis Jürgen Jost
title_fullStr	Postmodern analysis Jürgen Jost
title_full_unstemmed	Postmodern analysis Jürgen Jost
title_short	Postmodern analysis
title_sort	postmodern analysis
topic	Analisi matematica sbt Analysis Mathematical analysis Analysis (DE-588)4001865-9 gnd
topic_facet	Analisi matematica Analysis Mathematical analysis
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009852736&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT jostjurgen postmoderneanalysis AT jostjurgen postmodernanalysis

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