A ProblemText in Advanced Calculus
File : pdf, 2.7 MB, 381 pages
by John M. Erdman
TOC
Chapt 1. INTERVALS
1.1. DISTANCE AND NEIGHBORHOODS
1.2. INTERIOR OF A SET
Chapt 2. TOPOLOGY OF THE REAL LINE
2.1. OPEN SUBSETS OF R
2.2. CLOSED SUBSETS OF R
Chapt 3. CONTINUOUS FUNCTIONS FROM R TO R
3.1. CONTINUITY—AS A LOCAL PROPERTY
3.2. CONTINUITY—AS A GLOBAL PROPERTY
3.3. FUNCTIONS DEFINED ON SUBSETS OF R
Chapt 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
Chapt 5. CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM
5.1. CONNECTED SUBSETS OF R
5.2. CONTINUOUS IMAGES OF CONNECTED SETS
5.3. HOMEOMORPHISMS
Chapt 6. COMPACTNESS AND THE EXTREME VALUE THEOREM
6.1. COMPACTNESS
6.2. EXAMPLES OF COMPACT SUBSETS OF R
6.3. THE EXTREME VALUE THEOREM
Chapt 7. LIMITS OF REAL VALUED FUNCTIONS
7.1. DEFINITION
7.2. CONTINUITY AND LIMITS
Chapt 8. DIFFERENTIATION OF REAL VALUED FUNCTIONS
8.1. THE FAMILIES O AND o
8.2. TANGENCY
8.3. LINEAR APPROXIMATION
8.4. DIFFERENTIABILITY
Chapt 9. METRIC SPACES
9.1. DEFINITIONS
9.2. EXAMPLES
9.3. STRONGLY EQUIVALENT METRICS
Chapt 10. INTERIORS, CLOSURES, AND BOUNDARIES
10.1. DEFINITIONS AND EXAMPLES
10.2. INTERIOR POINTS
10.3. ACCUMULATION POINTS AND CLOSURES
Chapt 11. THE TOPOLOGY OF METRIC SPACES
11.1. OPEN AND CLOSED SETS
11.2. THE RELATIVE TOPOLOGY
Chapt 12. SEQUENCES IN METRIC SPACES
12.1. CONVERGENCE OF SEQUENCES
12.2. SEQUENTIAL CHARACTERIZATIONS OF TOPOLOGICAL PROPERTIES
12.3. PRODUCTS OF METRIC SPACES
Chapt 13. UNIFORM CONVERGENCE
13.1. THE UNIFORM METRIC ON THE SPACE OF BOUNDED FUNCTIONS
13.2. POINTWISE CONVERGENCE
Chapt 14. MORE ON CONTINUITY AND LIMITS
14.1. CONTINUOUS FUNCTIONS
14.2. MAPS INTO AND FROM PRODUCTS
14.3. LIMITS
Chapt 15. COMPACT METRIC SPACES
15.1. DEFINITION AND ELEMENTARY PROPERTIES
15.2. THE EXTREME VALUE THEOREM
15.3. DINI’S THEOREM
Chapt 16. SEQUENTIAL CHARACTERIZATION OF COMPACTNESS
16.1. SEQUENTIAL COMPACTNESS
16.2. CONDITIONS EQUIVALENT TO COMPACTNESS
16.3. PRODUCTS OF COMPACT SPACES
16.4. THE HEINE-BOREL THEOREM
Chapt 17. CONNECTEDNESS
17.1. CONNECTED SPACES
17.2. ARCWISE CONNECTED SPACES
Chapt 18. COMPLETE SPACES
18.1. CAUCHY SEQUENCES
18.2. COMPLETENESS
18.3. COMPLETENESS VS. COMPACTNESS
Chapt 19. APPLICATIONS OF A FIXED POINT THEOREM
19.1. THE CONTRACTIVE MAPPING THEOREM
19.2. APPLICATION TO INTEGRAL EQUATIONS
Chapt 20. VECTOR SPACES
20.1. DEFINITIONS AND EXAMPLES
20.2. LINEAR COMBINATIONS
20.3. CONVEX COMBINATIONS
Chapt 21. LINEARITY
21.1. LINEAR TRANSFORMATIONS
21.2. THE ALGEBRA OF LINEAR TRANSFORMATIONS
21.3. MATRICES
21.4. DETERMINANTS
21.5. MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS
Chapt 22. NORMS
22.1. NORMS ON LINEAR SPACES
22.2. NORMS INDUCE METRICS
22.3. PRODUCTS
22.4. THE SPACE B(S, V )
Chapt 23. CONTINUITY AND LINEARITY
23.1. BOUNDED LINEAR TRANSFORMATIONS
23.2. THE STONE-WEIERSTRASS THEOREM
23.3. BANACH SPACES
23.4. DUAL SPACES AND ADJOINTS
Chapt 24. THE CAUCHY INTEGRAL
24.1. UNIFORM CONTINUITY
24.2. THE INTEGRAL OF STEP FUNCTIONS
24.3. THE CAUCHY INTEGRAL
Chapt 25. DIFFERENTIAL CALCULUS
25.1. O AND o FUNCTIONS
25.2. TANGENCY
25.3. DIFFERENTIATION
25.4. DIFFERENTIATION OF CURVES
25.5. DIRECTIONAL DERIVATIVES
25.6. FUNCTIONS MAPPING INTO PRODUCT SPACES
Chapt 26. PARTIAL DERIVATIVES AND ITERATED INTEGRALS
26.1. THE MEAN VALUE THEOREM(S)
26.2. PARTIAL DERIVATIVES
26.3. ITERATED INTEGRALS
Chapt 27. COMPUTATIONS IN Rn
27.1. INNER PRODUCTS
27.2. THE GRADIENT
27.3. THE JACOBIAN MATRIX
27.4. THE CHAIN RULE
Chapt 28. INFINITE SERIES
28.1. CONVERGENCE OF SERIES
28.2. SERIES OF POSITIVE SCALARS
28.3. ABSOLUTE CONVERGENCE
28.4. POWER SERIES
Chapt 29. THE IMPLICIT FUNCTION THEOREM
29.1. THE INVERSE FUNCTION THEOREM
29.2. THE IMPLICIT FUNCTION THEOREM
Download : pdf1
