A ProblemText in Advanced Calculus

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A ProblemText in Advanced Calculus

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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

To Argentina

PREFACExi

FOR STUDENTS: HOW TO USE THIS PROBLEMTEXT

Chapter 1. INTERVALS

1.1. DISTANCE AND NEIGHBORHOODS

1.2. INTERIOR OF A SET

Chapter 2. TOPOLOGY OF THE REAL LINE

2.1. OPEN SUBSETS OFR5

2.2. CLOSED SUBSETS OFR7

Chapter 3. CONTINUOUS FUNCTIONS FROMRTOR9

3.1. CONTINUITY|AS A LOCAL PROPERTY

3.2. CONTINUITY|AS A GLOBAL PROPERTY

3.3. FUNCTIONS DEFINED ON SUBSETS OFR13

Chapter 4. SEQUENCES OF REAL NUMBERS

4.1. CONVERGENCE OF SEQUENCES

4.2. ALGEBRAIC COMBINATIONS OF SEQUENCES

4.3. SUFFICIENT CONDITION FOR CONVERGENCE

4.4. SUBSEQUENCES

23
Chapter 5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM 27

5.1. CONNECTED SUBSETS OFR27

5.2. CONTINUOUS IMAGES OF CONNECTED SETS

5.3. HOMEOMORPHISMS

30
Chapter 6. COMPACTNESS AND THE EXTREME VALUE THEOREM 33

6.1. COMPACTNESS

6.2. EXAMPLES OF COMPACT SUBSETS OFR34

6.3. THE EXTREME VALUE THEOREM

Chapter 7. LIMITS OF REAL VALUED FUNCTIONS

7.1. DEFINITION

7.2. CONTINUITY AND LIMITS

40
Chapter 8. DIFFERENTIATION OF REAL VALUED FUNCTIONS 43

8.1. THE FAMILIESOANDo43

8.2. TANGENCY

8.3. LINEAR APPROXIMATION

8.4. DIFFERENTIABILITY

Chapter 9. METRIC SPACES

9.1. DEFINITIONS

9.2. EXAMPLES

52
v vi CONTENTS

9.3. STRONGLY EQUIVALENT METRICS

Chapter 10. INTERIORS, CLOSURES, AND BOUNDARIES

10.1. DEFINITIONS AND EXAMPLES

10.2. INTERIOR POINTS

10.3. ACCUMULATION POINTS AND CLOSURES

Chapter 11. THE TOPOLOGY OF METRIC SPACES

11.1. OPEN AND CLOSED SETS

11.2. THE RELATIVE TOPOLOGY

Chapter 12. SEQUENCES IN METRIC SPACES

12.1. CONVERGENCE OF SEQUENCES

12.2. SEQUENTIAL CHARACTERIZATIONS OF TOPOLOGICAL PROPERTIES

12.3. PRODUCTS OF METRIC SPACES

Chapter 13. UNIFORM CONVERGENCE

13.1. THE UNIFORM METRIC ON THE SPACE OF BOUNDED FUNCTIONS

13.2. POINTWISE CONVERGENCE

Chapter 14. MORE ON CONTINUITY AND LIMITS

14.1. CONTINUOUS FUNCTIONS

14.2. MAPS INTO AND FROM PRODUCTS

14.3. LIMITS79

Chapter 15. COMPACT METRIC SPACES

15.1. DEFINITION AND ELEMENTARY PROPERTIES

15.2. THE EXTREME VALUE THEOREM

15.3. DINI'S THEOREM

85
Chapter 16. SEQUENTIAL CHARACTERIZATION OF COMPACTNESS 87

16.1. SEQUENTIAL COMPACTNESS

16.2. CONDITIONS EQUIVALENT TO COMPACTNESS

16.3. PRODUCTS OF COMPACT SPACES

16.4. THE HEINE-BOREL THEOREM

Chapter 17. CONNECTEDNESS

17.1. CONNECTED SPACES

17.2. ARCWISE CONNECTED SPACES

Chapter 18. COMPLETE SPACES

18.1. CAUCHY SEQUENCES

18.2. COMPLETENESS

18.3. COMPLETENESS VS. COMPACTNESS

Chapter 19. APPLICATIONS OF A FIXED POINT THEOREM

101

19.1. THE CONTRACTIVE MAPPING THEOREM

101

19.2. APPLICATION TO INTEGRAL EQUATIONS

105

Chapter 20. VECTOR SPACES

107

20.1. DEFINITIONS AND EXAMPLES

107

20.2. LINEAR COMBINATIONS

111

20.3. CONVEX COMBINATIONS

112

Chapter 21. LINEARITY

115

21.1. LINEAR TRANSFORMATIONS

115

CONTENTS vii

21.2. THE ALGEBRA OF LINEAR TRANSFORMATIONS

118

21.3. MATRICES

120

21.4. DETERMINANTS

124

21.5. MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS

125

Chapter 22. NORMS

129

22.1. NORMS ON LINEAR SPACES

129

22.2. NORMS INDUCE METRICS

130

22.3. PRODUCTS

131

22.4. THE SPACEB(S;V)134

Chapter 23. CONTINUITY AND LINEARITY

137

23.1. BOUNDED LINEAR TRANSFORMATIONS

137

23.2. THE STONE-WEIERSTRASS THEOREM

141

23.3. BANACH SPACES

143

23.4. DUAL SPACES AND ADJOINTS

144

Chapter 24. THE CAUCHY INTEGRAL

145

24.1. UNIFORM CONTINUITY

145

24.2. THE INTEGRAL OF STEP FUNCTIONS

147

24.3. THE CAUCHY INTEGRAL

150

Chapter 25. DIFFERENTIAL CALCULUS

157

25.1.OANDoFUNCTIONS157

25.2. TANGENCY

159

25.3. DIFFERENTIATION

160

25.4. DIFFERENTIATION OF CURVES

163

25.5. DIRECTIONAL DERIVATIVES

165

25.6. FUNCTIONS MAPPING INTO PRODUCT SPACES

166
Chapter 26. PARTIAL DERIVATIVES AND ITERATED INTEGRALS 169

26.1. THE MEAN VALUE THEOREM(S)

169

26.2. PARTIAL DERIVATIVES

173

26.3. ITERATED INTEGRALS

177

Chapter 27. COMPUTATIONS INRn181

27.1. INNER PRODUCTS

181

27.2. THE GRADIENT

183

27.3. THE JACOBIAN MATRIX

187

27.4. THE CHAIN RULE

188

Chapter 28. INFINITE SERIES

195

28.1. CONVERGENCE OF SERIES

195

28.2. SERIES OF POSITIVE SCALARS

200

28.3. ABSOLUTE CONVERGENCE

200

28.4. POWER SERIES

202

Chapter 29. THE IMPLICIT FUNCTION THEOREM

209

29.1. THE INVERSE FUNCTION THEOREM

209

29.2. THE IMPLICIT FUNCTION THEOREM

213

Appendix A. QUANTIFIERS

219

Appendix B. SETS

221
viii CONTENTS

Appendix C. SPECIAL SUBSETS OFR225

Appendix D. LOGICAL CONNECTIVES

227

D.1. DISJUNCTION AND CONJUNCTION

227

D.2. IMPLICATION

228

D.3. RESTRICTED QUANTIFIERS

229

D.4. NEGATION

230

Appendix E. WRITING MATHEMATICS

233

E.1. PROVING THEOREMS

233

E.2. CHECKLIST FOR WRITING MATHEMATICS

234

E.3. FRAKTUR AND GREEK ALPHABETS

236

Appendix F. SET OPERATIONS

237

F.1. UNIONS237

F.2. INTERSECTIONS

239

F.3. COMPLEMENTS

240

Appendix G. ARITHMETIC

243
quotesdbs_dbs29.pdfusesText_35

[PDF] A ProblemText in Advanced Calculus

A ProblemText in Advanced Calculus

John M. Erdman

Portland State University

Version July 1, 2014

E-mail address:erdman@pdx.edu

To Argentina

Contents

PREFACExi

FOR STUDENTS: HOW TO USE THIS PROBLEMTEXT

Chapter 1. INTERVALS

1.1. DISTANCE AND NEIGHBORHOODS

1.2. INTERIOR OF A SET

Chapter 2. TOPOLOGY OF THE REAL LINE

2.1. OPEN SUBSETS OFR5

2.2. CLOSED SUBSETS OFR7

Chapter 3. CONTINUOUS FUNCTIONS FROMRTOR9

3.1. CONTINUITY|AS A LOCAL PROPERTY

3.2. CONTINUITY|AS A GLOBAL PROPERTY

3.3. FUNCTIONS DEFINED ON SUBSETS OFR13

Chapter 4. SEQUENCES OF REAL NUMBERS

4.1. CONVERGENCE OF SEQUENCES

4.2. ALGEBRAIC COMBINATIONS OF SEQUENCES

4.3. SUFFICIENT CONDITION FOR CONVERGENCE

4.4. SUBSEQUENCES

5.1. CONNECTED SUBSETS OFR27

5.2. CONTINUOUS IMAGES OF CONNECTED SETS

5.3. HOMEOMORPHISMS

6.1. COMPACTNESS

6.2. EXAMPLES OF COMPACT SUBSETS OFR34

6.3. THE EXTREME VALUE THEOREM

Chapter 7. LIMITS OF REAL VALUED FUNCTIONS

7.1. DEFINITION

7.2. CONTINUITY AND LIMITS

8.1. THE FAMILIESOANDo43

8.2. TANGENCY

8.3. LINEAR APPROXIMATION

8.4. DIFFERENTIABILITY

Chapter 9. METRIC SPACES

9.1. DEFINITIONS

9.2. EXAMPLES

9.3. STRONGLY EQUIVALENT METRICS

Chapter 10. INTERIORS, CLOSURES, AND BOUNDARIES

10.1. DEFINITIONS AND EXAMPLES

10.2. INTERIOR POINTS

10.3. ACCUMULATION POINTS AND CLOSURES

Chapter 11. THE TOPOLOGY OF METRIC SPACES

11.1. OPEN AND CLOSED SETS

11.2. THE RELATIVE TOPOLOGY

Chapter 12. SEQUENCES IN METRIC SPACES

12.1. CONVERGENCE OF SEQUENCES

12.2. SEQUENTIAL CHARACTERIZATIONS OF TOPOLOGICAL PROPERTIES

12.3. PRODUCTS OF METRIC SPACES

Chapter 13. UNIFORM CONVERGENCE

13.1. THE UNIFORM METRIC ON THE SPACE OF BOUNDED FUNCTIONS

13.2. POINTWISE CONVERGENCE

Chapter 14. MORE ON CONTINUITY AND LIMITS

14.1. CONTINUOUS FUNCTIONS

14.2. MAPS INTO AND FROM PRODUCTS

14.3. LIMITS79

Chapter 15. COMPACT METRIC SPACES

15.1. DEFINITION AND ELEMENTARY PROPERTIES

15.2. THE EXTREME VALUE THEOREM

15.3. DINI'S THEOREM

16.1. SEQUENTIAL COMPACTNESS

16.2. CONDITIONS EQUIVALENT TO COMPACTNESS

16.3. PRODUCTS OF COMPACT SPACES

16.4. THE HEINE-BOREL THEOREM

Chapter 17. CONNECTEDNESS

17.1. CONNECTED SPACES

17.2. ARCWISE CONNECTED SPACES

Chapter 18. COMPLETE SPACES

18.1. CAUCHY SEQUENCES

18.2. COMPLETENESS

18.3. COMPLETENESS VS. COMPACTNESS

Chapter 19. APPLICATIONS OF A FIXED POINT THEOREM

19.1. THE CONTRACTIVE MAPPING THEOREM

19.2. APPLICATION TO INTEGRAL EQUATIONS

Chapter 20. VECTOR SPACES

20.1. DEFINITIONS AND EXAMPLES