A problems based course in advanced calculus

✍ Scribed by Erdman, John M

• Publisher: American Mathematical Society
• Year: 2018
• Tongue: English
• Leaves: 384
• Series: Sally series (Providence R.I.); Pure and applied undergraduate texts 32
• Category: Library

✦ Table of Contents

Content: Machine generated contents note: 1.1.Distance and neighborhoods --
1.2.Interior of a set --
2.1.Open subsets of Real Numbers --
2.2.Closed subsets of Real Numbers --
3.1.Continuity-as a local property --
3.2.Continuity-as a global property --
3.3.Functions defined on subsets 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 of Real Numbers --
5.2.Continuous images of connected sets --
5.3.Homeomorphisms --
6.1.Compactness --
6.2.Examples of compact subsets of Real Numbers --
6.3.The extreme value theorem --
7.1.Definition --
7.2.Continuity and limits --
8.1.The families of "big-oh" and "little-oh" --
8.2.Tangency --
8.3.Linear approximation --
8.4.Differentiability --
9.1.Definitions --
9.2.Examples --
9.3.Equivalent metrics --
10.1.Definitions and examples --
10.2.Interior points --
10.3.Accumulation points and closures --
11.1.Open and closed sets Note continued: 11.2.The relative topology --
12.1.Convergence of sequences --
12.2.Sequential characterizations of topological properties --
12.3.Products of metric spaces --
13.1.The uniform metric on the space of bounded functions --
13.2.Pointwise convergence --
14.1.Continuous functions --
14.2.Maps into and from products --
14.3.Limits --
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 --
17.1.Connected spaces --
17.2.Arcwise connected spaces --
18.1.Cauchy sequences --
18.2.Completeness --
18.3.Completeness vs. compactness --
19.1.The contractive mapping theorem --
19.2.Application to integral equations --
20.1.Definitions and examples --
20.2.Linear combinations --
20.3.Convex combinations --
21.1.Linear transformations --
21.2.The algebra of linear transformations Note continued: 21.3.Matrices --
21.4.Determinants --
21.5.Matrix representations of linear transformations --
22.1.Norms on linear spaces --
22.2.Norms induce metrics --
22.3.Products --
22.4.The space B(S,V) --
23.1.Bounded linear transformations --
23.2.The Stone-Weierstrass theorem --
23.3.Banach spaces --
23.4.Dual spaces and adjoints --
24.1.Uniform continuity --
24.2.The integral of step functions --
24.3.The Cauchy integral --
25.1."Big-oh" and "little-oh" functions --
25.2.Tangency --
25.3.Differentiation --
25.4.Differentiation of curves --
25.5.Directional derivatives --
25.6.Functions mapping into product spaces --
26.1.The mean value theorem(s) --
26.2.Partial derivatives --
26.3.Iterated integrals --
27.1.Inner products --
27.2.The gradient --
27.3.The Jacobian matrix --
27.4.The chain rule --
28.1.Convergence of series --
28.2.Series of positive scalars --
28.3.Absolute convergence --
28.4.Power series --
29.1.The inverse function theorem Note continued: 29.2.The implicit function theorem --
30.1.Multilinear functions --
30.2.Second order differentials --
30.3.Higher order differentials --
D.1.Disjunction and conjunction --
D.2.Implication --
D.3.Restricted quantifiers --
D.4.Negation --
E.1.Proving theorems --
E.2.Checklist for writing mathematics --
E.3.Fraktur and Greek alphabets --
F.1.Unions --
F.2.Intersections --
F.3.Complements --
G.1.The field axioms --
G.2.Uniqueness of identities --
G.3.Uniqueness of inverses --
G.4.Another consequence of uniqueness --
J.1.Upper and lower bounds --
J.2.Least upper and greatest lower bounds --
J.3.The least upper bound axiom for Real Numbers --
J.4.The Archimedean property --
K.1.Cartesian products --
K.2.Relations --
K.3.Functions --
L.1.Images and inverse images --
L.2.Composition of functions --
L.3.The identity function --
L.4.Diagrams --
L.5.Restrictions and extensions --
M.1.Injections, surjections, and bijections --
M.2.Inverse functions.

✦ Subjects

Calculus -- Textbooks.;Calculus -- Study and teaching (Graduate);General -- General and miscellaneous specific topics -- Problem books.;General -- Instructional exposition (textbooks, tutorial papers, etc.).;Calculus.