Advanced calculus:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Pure and applied undergraduate texts
5 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke. - Originally published: 2nd ed. Belmont, CA : Thomson Brooks/Cole, c2006. -- Includes index. |
| Beschreibung: | xviii, 590 Seiten Illustrationen 24 cm |
| ISBN: | 9780821847916 0821847910 |
Internformat
MARC
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| 100 | 1 | |a Fitzpatrick, Patrick |d 1946- |0 (DE-588)14081888X |4 aut | |
| 245 | 1 | 0 | |a Advanced calculus |c Patrick M. Fitzpatrick |
| 250 | |a Second edition | ||
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2006] | |
| 264 | 4 | |c © 2006 | |
| 300 | |a xviii, 590 Seiten |b Illustrationen |c 24 cm | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Pure and applied undergraduate texts |v 5 | |
| 490 | 0 | |a The Sally series | |
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Datensatz im Suchindex
| _version_ | 1829099957974466560 |
|---|---|
| adam_text |
CONTENTS
Preface
xi
Preliminaries t
1
TOOLS FOR ANALYSIS
5
1.1
The Completeness Axiom and Some of Its Consequences
5
1.2
The Distribution of the Integers and the Rational Numbers
12
1.3
Inequalities and Identities
16
CONVERGENT SEQUENCES
23
2.1
The Convergence of Sequences
23
2.2
Sequences and Sets
35
2.3
The Monotone Convergence Theorem
38
2.4
The Sequential Compactness Theorem
43
2.5
Covering Properties of Sets*
47
CONTINUOUS FUNCTIONS
53
3.Ί
Continuity
53
3.2
The Extreme Value Theorem
58
3.3
The Intermediate Value Theorem
62
3.4
Uniform Continuity
66
3.5
The 6-<5 Criterion for Continuity
70
3.6
Images and Inverses; Monotone Functions
74
3.7
Limits
81
DIFFERENTIATION
87
4.1
The Algebra of Derivatives
87
4.2
Differentiating Inverses and Compositions
96
4.3
The Mean Value Theorem and Its Geometric Consequences
101
4.4
The Cauchy Mean Value Theorem and Its Analytic Consequences
111
4.5
The Notation of Leibnitz
113
VI
CONTENTS
5
ELEMENTARY FUNCTIONS AS SOLUTIONS
OF DIFFERENTIAL EQUATIONS
116
5.1
Solutions of Differential Equations
116
5.2
The Natural Logarithm and Exponential Functions
118
5.3
The Trigonometric Functions
125
5.4
The Inverse Trigonometric Functions
132
INTEGRATION: TWO FUNDAMENTAL THEOREMS
135
6.1
Darboux Sums; Upper and Lower Integrals
135
6.2
The Archimedes-Riemann Theorem
142
6.3
Additivity,
Monotonicity,
and Linearity
150
6.4
Continuity and Integrability
155
6.5
The First Fundamental Theorem: Integrating Derivatives
160
6.6
The Second Fundamental Theorem: Differentiating Integrals
165
T
_
INTEGRATION: FURTHER TOPICS
175
7.1
Solutions of Differential Equations
1 75
7.2
Integration by Parts and by Substitution
178
7.3
The Convergence of Darboux and Riemann Sums
183
7.4
The Approximation of Integrals
190
8_
APPROXIMATION BY TAYLOR POLYNOMIALS
199
8.1
Taylor Polynomials
199
8.2
The
Lagrange
Remainder Theorem
203
8.3
The Convergence of Taylor Polynomials
209
8.4
A Power Series for the Logarithm
212
S3
The Cauchy Integral Remainder Theorem
215
8.6
A Nonanalytic, Infinitely Differentiable Function
221
8.7
The
Weierstrass
Approximation Theorem
223
SEQUENCES AND SERIES OF FUNCTIONS
228
9.1
Sequences and Series of Numbers
228
9.2
Poi
ntwise Convergence of Sequences of Functions
241
CONTENTS
Vil
9.3 Uniform
Convergence
of Sequences of Functions
245
9.4
The Uniform Limit of Functions
249
9.5
Power Series
255
9.6
A Continuous Nowhere
Differenti
able Function
264
10_
THE EUCLIDEAN SPACE Rn
269
10.1
The Linear Structure of
Џп
and the Scalar Product
269
10.2
Convergence of Sequences in Rn
277
10.3
Open Sets and Closed Sets in Rn
282
1!_
CONTINUITY, COMPACTNESS, AND CONNECTEDNESS
290
11.1
Continuous Functions and Mappings
290
11.2
Sequential Compactness, Extreme Values,
and Uniform Continuity
298
11.3
Rathwise Connectedness and the Intermediate Value Theorem*
304
11.4
Connectedness and the Intermediate Value Property*
310
METRIC SPACES
314
12.1
Open Sets, Closed Sets, and Sequential Convergence
314
12.2
Completeness and the Contraction Mapping Principle
322
12.3
The Existence Theorem for Nonlinear Differential Equations
328
12.4
Continuous Mappings between Metric Spaces
337
12.5
Sequential Compactness and Connectedness
342
13_
DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES
348
13.1
Limits
348
13.2
Partial Derivatives
353
13.3
The Mean Value Theorem and Directional Derivatives
364
14_
LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS
372
14.1
First-Order Approximation, Tangent Planes, and
Affine
Functions
372
14.2
Quadratic Functions, Hessian Matrices, and Second Derivatives*
380
14.3
Second-Order Approximation and the Second-Derivative Test*
387
Vlil
CONTENTS
15
APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS
394
15.1
Linear Mappings and Matrices
394
15.2
The Derivative Matrix and the Differential
407
15.3
The Chain Rule
414
16_
IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM
421
16.1
Functions of a Single Variable and Maps in the Plane
421
16.2
Stability of Nonlinear Mappings
429
16.3
A Minimization Principle and the General
Inverse Function Theorem
433
17_
THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS
440
17.1
A Scalar Equation in Two Unknowns: Dini's Theorem
440
17.2
The General Implicit Function Theorem
449
17.3
Equations of Surfaces and Paths in
Ж3
454
17.4
Constrained
Extrema
Problems and
Lagrange
Multipliers
460
18_
INTEGRATING FUNCTIONS OF SEVERAL VARIABLES
470
18.1
Integration of Functions on Generalized Rectangles
470
18.2
Continuity and Integrability
482
18.3
Integration of Functions on Jordan Domains
489
19_
ITERATED INTEGRATION AND CHANGES OF VARIABLES
498
19.1
Fubini's Theorem
498
19.2
The Change of Variables Theorem: Statements and Examples
505
19.3
Proof of the Change of Variables Theorem
510
20_
LINE AND SURFACE INTEGRALS
520
20.1
Arclength and Line Integrals
520
20.2
Surface Area and Surface Integrals
533
20.3
The Integral Formulas of Green and Stokes
543
CONTENTS
IX
CONSEQUENCES
OF THE FIELD AND
POSITIVITY
AXIOMS
559
A.1 The Field Axioms and Their Consequences
559
A.2 The
Positivity
Axioms and Their Consequences
563
В
_
LINEAR ALGEBRA
565
Index
581
Advanced
Calculus is intended as a text for courses that furnish the backbone of
the student's undergraduate education in mathematical analysis. The goal is to
rigorously present the fundamental concepts within the context of illuminating
examples and stimulating exercises. This book is self-contained and starts with
the creation of basic tools using the completeness axiom. The continuity, differ¬
entiability, integrability, and power series representation properties of functions
of a single variable are established. The next few chapters describe the
topolog-
ical and metric properties of Euclidean space. These are the basis of a rigorous
treatment of differential calculus (including the Implicit Function Theorem and
Lagrange
Multipliers) for mappings between Euclidean spaces and integration
for functions of several real variables.
Special attention has been paid to the motivation for proofs. Selected topics,
such as the
Picard
Existence Theorem for differential equations, have been
included in such a way that selections may be made while preserving a fluid
presentation of the essential material.
Supplemented with numerous exercises, Advanced Calculus is a perfect book
for undergraduate students of analysis. |
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| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Second edition |
| format | Book |
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| indexdate | 2025-04-11T10:00:51Z |
| institution | BVB |
| isbn | 9780821847916 0821847910 |
| language | English |
| lccn | 2008047395 |
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| physical | xviii, 590 Seiten Illustrationen 24 cm |
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| series | Pure and applied undergraduate texts |
| series2 | Pure and applied undergraduate texts The Sally series |
| spelling | Fitzpatrick, Patrick 1946- (DE-588)14081888X aut Advanced calculus Patrick M. Fitzpatrick Second edition Providence, Rhode Island American Mathematical Society [2006] © 2006 xviii, 590 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 5 The Sally series Hier auch später erschienene, unveränderte Nachdrucke. - Originally published: 2nd ed. Belmont, CA : Thomson Brooks/Cole, c2006. -- Includes index. Calculus Textbooks Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Analysis (DE-588)4001865-9 s DE-604 Pure and applied undergraduate texts 5 (DE-604)BV035489189 5 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026142641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026142641&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Fitzpatrick, Patrick 1946- Advanced calculus Pure and applied undergraduate texts Calculus Textbooks Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4123623-3 |
| title | Advanced calculus |
| title_auth | Advanced calculus |
| title_exact_search | Advanced calculus |
| title_full | Advanced calculus Patrick M. Fitzpatrick |
| title_fullStr | Advanced calculus Patrick M. Fitzpatrick |
| title_full_unstemmed | Advanced calculus Patrick M. Fitzpatrick |
| title_short | Advanced calculus |
| title_sort | advanced calculus |
| topic | Calculus Textbooks Analysis (DE-588)4001865-9 gnd |
| topic_facet | Calculus Textbooks Analysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026142641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026142641&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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