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Schuams Outline of Theory and Problems in Advanced Calculus
Advanced calculus is not a single theory. However the various sub-theories
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(1) to obtain a body of knowledge in Advanced Calculus the basis of the analysis of real-valued functions of one real variable;. (2) to learn how to
ADVANCED CALCULUS I & II Contents References 4 Index 5 Part 1
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The course in advanced calculus contained in this book has for many years been given by the author to students in the Massa- chusetts Institute of
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Many of the ideas of advanced calculus are by no means simple or intuitive. (It took mathematicians two hundred years to develop them!) Furthermore for
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In Chapter 8 certain basic concepts associated with the simpler types of partial differential equations are introduced
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L Y N N H. L 0 0 MIS and S H L 0 M 0 S T ERN B ERG. Department of Mathematics Harvard University. ADVANCED CALCULUS. REVISED EDITION.
Schuams Outline of Theory and Problems in Advanced Calculus
Advanced calculus is not a single theory. However the various sub-theories
ADVANCED CALCULUS - Lecture notes for MA 440/540 & 441/541
(1) to obtain a body of knowledge in Advanced Calculus the basis of the analysis of http://www.math.uab.edu/~rudi/teaching/algebra.pdf ...
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PREFACE The course in advanced calculus contained in this book has for many years been given by the author to students in the Massa-
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In conducting this class we shall try to model a mathematical community in which both collaboration and competition are prevalent This community is — no you
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A ProblemText in Advanced Calculus
John M. Erdman
Portland State University
Version July 1, 2014
E-mail address:erdman@pdx.edu
iiTo Argentina
Contents
PREFACExi
FOR STUDENTS: HOW TO USE THIS PROBLEMTEXT
xvChapter 1. INTERVALS
11.1. DISTANCE AND NEIGHBORHOODS
11.2. INTERIOR OF A SET
2Chapter 2. TOPOLOGY OF THE REAL LINE
52.1. OPEN SUBSETS OFR5
2.2. CLOSED SUBSETS OFR7
Chapter 3. CONTINUOUS FUNCTIONS FROMRTOR9
3.1. CONTINUITY|AS A LOCAL PROPERTY
93.2. CONTINUITY|AS A GLOBAL PROPERTY
103.3. FUNCTIONS DEFINED ON SUBSETS OFR13
Chapter 4. SEQUENCES OF REAL NUMBERS
174.1. CONVERGENCE OF SEQUENCES
174.2. ALGEBRAIC COMBINATIONS OF SEQUENCES
194.3. SUFFICIENT CONDITION FOR CONVERGENCE
204.4. SUBSEQUENCES
23Chapter 5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM 27
5.1. CONNECTED SUBSETS OFR27
5.2. CONTINUOUS IMAGES OF CONNECTED SETS
295.3. HOMEOMORPHISMS
30Chapter 6. COMPACTNESS AND THE EXTREME VALUE THEOREM 33
6.1. COMPACTNESS
336.2. EXAMPLES OF COMPACT SUBSETS OFR34
6.3. THE EXTREME VALUE THEOREM
36Chapter 7. LIMITS OF REAL VALUED FUNCTIONS
397.1. DEFINITION
397.2. CONTINUITY AND LIMITS
40Chapter 8. DIFFERENTIATION OF REAL VALUED FUNCTIONS 43
8.1. THE FAMILIESOANDo43
8.2. TANGENCY
458.3. LINEAR APPROXIMATION
468.4. DIFFERENTIABILITY
47Chapter 9. METRIC SPACES
519.1. DEFINITIONS
519.2. EXAMPLES
52v vi CONTENTS
9.3. STRONGLY EQUIVALENT METRICS
55Chapter 10. INTERIORS, CLOSURES, AND BOUNDARIES
5710.1. DEFINITIONS AND EXAMPLES
5710.2. INTERIOR POINTS
5810.3. ACCUMULATION POINTS AND CLOSURES
58Chapter 11. THE TOPOLOGY OF METRIC SPACES
6111.1. OPEN AND CLOSED SETS
6111.2. THE RELATIVE TOPOLOGY
63Chapter 12. SEQUENCES IN METRIC SPACES
6512.1. CONVERGENCE OF SEQUENCES
6512.2. SEQUENTIAL CHARACTERIZATIONS OF TOPOLOGICAL PROPERTIES
6512.3. PRODUCTS OF METRIC SPACES
66Chapter 13. UNIFORM CONVERGENCE
6913.1. THE UNIFORM METRIC ON THE SPACE OF BOUNDED FUNCTIONS
6913.2. POINTWISE CONVERGENCE
70Chapter 14. MORE ON CONTINUITY AND LIMITS
7314.1. CONTINUOUS FUNCTIONS
7314.2. MAPS INTO AND FROM PRODUCTS
7714.3. LIMITS79
Chapter 15. COMPACT METRIC SPACES
8315.1. DEFINITION AND ELEMENTARY PROPERTIES
8315.2. THE EXTREME VALUE THEOREM
8415.3. DINI'S THEOREM
85Chapter 16. SEQUENTIAL CHARACTERIZATION OF COMPACTNESS 87
16.1. SEQUENTIAL COMPACTNESS
8716.2. CONDITIONS EQUIVALENT TO COMPACTNESS
8816.3. PRODUCTS OF COMPACT SPACES
8916.4. THE HEINE-BOREL THEOREM
90Chapter 17. CONNECTEDNESS
9317.1. CONNECTED SPACES
9317.2. ARCWISE CONNECTED SPACES
94Chapter 18. COMPLETE SPACES
9718.1. CAUCHY SEQUENCES
9718.2. COMPLETENESS
9718.3. COMPLETENESS VS. COMPACTNESS
98Chapter 19. APPLICATIONS OF A FIXED POINT THEOREM
10119.1. THE CONTRACTIVE MAPPING THEOREM
10119.2. APPLICATION TO INTEGRAL EQUATIONS
105Chapter 20. VECTOR SPACES
10720.1. DEFINITIONS AND EXAMPLES
10720.2. LINEAR COMBINATIONS
11120.3. CONVEX COMBINATIONS
112Chapter 21. LINEARITY
11521.1. LINEAR TRANSFORMATIONS
115CONTENTS vii
21.2. THE ALGEBRA OF LINEAR TRANSFORMATIONS
11821.3. MATRICES
12021.4. DETERMINANTS
12421.5. MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS
125Chapter 22. NORMS
12922.1. NORMS ON LINEAR SPACES
12922.2. NORMS INDUCE METRICS
13022.3. PRODUCTS
13122.4. THE SPACEB(S;V)134
Chapter 23. CONTINUITY AND LINEARITY
13723.1. BOUNDED LINEAR TRANSFORMATIONS
13723.2. THE STONE-WEIERSTRASS THEOREM
14123.3. BANACH SPACES
14323.4. DUAL SPACES AND ADJOINTS
144Chapter 24. THE CAUCHY INTEGRAL
14524.1. UNIFORM CONTINUITY
14524.2. THE INTEGRAL OF STEP FUNCTIONS
14724.3. THE CAUCHY INTEGRAL
150Chapter 25. DIFFERENTIAL CALCULUS
15725.1.OANDoFUNCTIONS157
25.2. TANGENCY
15925.3. DIFFERENTIATION
16025.4. DIFFERENTIATION OF CURVES
16325.5. DIRECTIONAL DERIVATIVES
16525.6. FUNCTIONS MAPPING INTO PRODUCT SPACES
166Chapter 26. PARTIAL DERIVATIVES AND ITERATED INTEGRALS 169
26.1. THE MEAN VALUE THEOREM(S)
16926.2. PARTIAL DERIVATIVES
17326.3. ITERATED INTEGRALS
177Chapter 27. COMPUTATIONS INRn181
27.1. INNER PRODUCTS
18127.2. THE GRADIENT
18327.3. THE JACOBIAN MATRIX
18727.4. THE CHAIN RULE
188Chapter 28. INFINITE SERIES
19528.1. CONVERGENCE OF SERIES
19528.2. SERIES OF POSITIVE SCALARS
20028.3. ABSOLUTE CONVERGENCE
20028.4. POWER SERIES
202Chapter 29. THE IMPLICIT FUNCTION THEOREM
20929.1. THE INVERSE FUNCTION THEOREM
20929.2. THE IMPLICIT FUNCTION THEOREM
213Appendix A. QUANTIFIERS
219Appendix B. SETS
221viii CONTENTS
Appendix C. SPECIAL SUBSETS OFR225
Appendix D. LOGICAL CONNECTIVES
227D.1. DISJUNCTION AND CONJUNCTION
227D.2. IMPLICATION
228D.3. RESTRICTED QUANTIFIERS
229D.4. NEGATION
230Appendix E. WRITING MATHEMATICS
233E.1. PROVING THEOREMS
233E.2. CHECKLIST FOR WRITING MATHEMATICS
234E.3. FRAKTUR AND GREEK ALPHABETS
236Appendix F. SET OPERATIONS
237F.1. UNIONS237
F.2. INTERSECTIONS
239F.3. COMPLEMENTS
240Appendix G. ARITHMETIC
243quotesdbs_dbs29.pdfusesText_35
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