5/12/15, 1:30-3:30 p.m.: Final. [Text: Chapter 6 (6.1-6, 6.8,
6.12(a-d), 6.20-22), the Riemann sums material from your class notes,
Chapter 7, Chapter 8 (pp. 172-174, 178-185), Chapter 9 (9.4, 9.6(a,c),
9.10-18, 20-21, 24-29, 39, 41), and Chapter 11 (11.1-8, 11.10-25,
28-30)]
5/11/15, 1:00-1:50 p.m.: Extra office hours.
5/8/15, 1:30-2:20 p.m.: Extra office hours.
5/8/15: Discussion of the Sample Final (selected problems). [Text:
Chapter 6 (6.1-6, 6.8, 6.12(a-d), 6.20-22), the Riemann sums material
from your class notes, Chapter 7, Chapter 8 (pp. 172-174, 178-185),
Chapter 9 (9.4, 9.6(a,c), 9.10-18, 20-21, 24-29, 39, 41), and Chapter
11 (11.1-8, 11.10-25, 28-30)]
5/8/15, 11:15 a.m. - 12:05 p.m.: Regular office hours.
5/6/15: The case of f = lim fn = ∞ on a set of
positive measure in the Monotone Convergence Theorem. Revisiting the
Monotone Convergence Theorem. Using bounded and unbounded simple
functions in the definition of Lebesgue integral gives the same
result. [Text: 11.21, 28]
5/4/15: Discussion of Midterm II. [Text: Chapter 9 (9.15-18, 20-21,
24-29, 39, 41), Chapter 11 (11.1-8, 11.10-25, 28-30)]
5/1/15: Midterm II. [Text: Chapter 9 (9.15-18, 20-21, 24-29,
39, 41), Chapter 11 (11.1-8, 11.10-25, 28-30)]
4/29/15: Review for Midterm II: discussion of the sample test. [Text:
Chapter 9 (9.15-18, 20-21, 24-29, 39, 41), Chapter 11 (11.1-8,
11.10-25, 28-30)]
4/27/15: Countable additivity of the integral of a simple
function. Lebesgue's monotone convergence theorem. Additivity of the
integral. Integrals of not necessarily negative functions. [Text:
11.28-30, 24, 22]
4/24/15: Homework is due at the beginning of Friday class. Discussion
of homework (Problem 1). Example: Lebesgue vs. Riemann. [Text:
11.13-25]
4/22/15: The Lebesgue integral. First properties. [Text: 11.21-25]
4/20/15: Measurable functions. Simple functions. [Text: 11.13-20]
4/17/15: Homework is due at the beginning of Friday class. An example
of a Lebesgue non-measurable set. Discussion of homework. Measurable
functions: comments on definition. [Text: 11.8, 11.10-11, 11.13]
4/15/15: Finishing the proof of the theorem on measurable sets forming
a σ-algebra and the outer measure restricting to a
measure. Completeness of Lebesgue measure. Lebesgue measurable sets
are almost Borel. [Text: 11.11]
4/13/15: Proofs of theorems from previous class meetings: a rectangle
is Lebesgue measurable; Borel sets are Lebesgue measurable; measurable
sets with respect to an outer measure form a σ-algebra, and the
outer measure on measurable sets is a measure (a sketch of
proof). [Text: 11.8, 11.10-11]
4/10/15: Homework is due at the beginning of Friday class. Discussion
of homework Problem 5: regularity of Lebesgue measure. [Text: 11.8,
11.10-11]
4/8/15: The Lebesgue outer measure of a rectangle (finished). Proof of
countable subadditivity of the Lebesgue outer measure. Rectangles and
their unions are Lebesgue measurable sets (no proof). The
σ-algebra of measurable sets (no proof). [Text: 11.10-11]
4/6/15: A recap on measures, outer measures, and the Lebesgue versions
of them in Rn. Lebesgue measurable sets. The
Lebesgue measure of a Lebesgue measurable set. The Lebesgue outer
measure of a rectangle (not finished). [Text: 11.8]
4/3/15: Homework is due at the beginning of Friday class. Discussion
of homework. Measurable sets. The measure of a measurable set. [Text:
11.1-8, 11.12]
4/1/15: The Banach-Tarski paradox. Measures, measure
spaces. Properties of measures. The Lebesgue outer measure. An
abstract outer measure. [Text: 11.4-8]
3/30/15: Sigma-algebras and measures. [Text: 11.1-3, 11.12]
3/27/15: Homework is due at the beginning of Friday class. Discussion
of homework. Finishing the proof of the implicit function theorem for
real-valued functions. [Text: Chapter 9: pp. 221-228]
3/25/15: The implicit function theorem: m = 1 (the case of real-valued
functions). The inverse function theorem. [Text: Chapter 9:
pp. 221-228]
3/23/15: Homework is due at the beginning of Monday class. Discussion
of homework. Idea of an implicit function. [Text: 9.15-18, 20-21, 39,
41, Chapter 9: pp. 223-224]
3/16-20/15: Spring Break. [Text: 9.15-18, 20-21, 39, 41]
3/14/15: The pi day: 3/14/15.
3/13/15: Continuity of partial derivatives and
differentiability. Higher-order partial derivatives. [Text: 9.20-21,
39, 41]
3/11/15: The chain rule. Partial derivatives and the total
derivative. The gradient. [Text: 9.15-18]
3/9/15: Discussion of Midterm I. Directional derivative. The sum and
product rules. [Text: 9.18]
3/6/15: Midterm I. [Text: Chapter 6 (6.1-6, 6.8, 6.12(a-d),
6.20-22), the Riemann sums material from your class notes, Chapter 7,
Chapter 8 (pp. 172-174, 178-185), and Chapter 9 (9.4, 9.6(a,c),
9.10-14)]
3/4/15: Review for Midterm I: discussion of the sample test. [Text:
Chapter 6 (6.1-6, 6.8, 6.12(a-d), 6.20-22), the Riemann sums material
from your class notes, Chapter 7, Chapter 8 (pp. 172-174, 178-185),
and Chapter 9 (9.4, 9.6(a,c), 9.10-14)]
3/2/15: The notion of the derivative of a function of several
variables. [Text: Chapter 9: 9.4, 9.6(a,c), 9.10-14]
2/27/15: The Fundamental Theorem of Algebra. Homework is due at the
beginning of the Friday class. Discussion of homework. [Text: 7.10,
Chapter 8: pp. 172-174, 178-181, 182-185]
2/25/15: The exponential function and the logarithm. Trigonometric
functions. [Text: Chapter 8: pp. 178-184]
2/23/15: Uniform convergence and differentiation of power series. The
exponential function. [Text: 7.10, Chapter 8 up to 8.2 (not
including), p. 178]
2/20/15: The sketch of proof of the Arzelà-Ascoli
theorem. Homework is due at the beginning of the Friday
class. Discussion of homework. [Text: 7.3-6, 7.16-25]
2/18/15: Differentiating uniform limits. Equicontinuous families and
the Arzelà-Ascoli theorem (without proof). [Text: 7.17, 19-25]
2/16/15: Examples of pointwise limits not commuting with
differentiation and integration. Uniform limits in the integral and
derivative. [Text: 7.3-6, 7.16]
2/13/15: Homework is due at the beginning of the Friday
class. Discussion of homework. [Text: 7.26-32]
2/11/15: The proof of the Stone theorem; lattices, Lemma 2, and the end
of the proof. [Text: 7.31-32, Steps 3-4.]
2/9/15: The proof of the Stone theorem: Lemma 1. [Text: 7.26-30, 32,
Steps 1-2]
2/6/15: Homework is due at the beginning of the Friday
class. Discussion of homework. The proof of Dini's theorem. [Text:
Chapter 7 through 7.15, and 7.26-33 (Skip proofs in 7.31-33 for the
time being)]
2/4/15: Completeness of the subspace of continuous functions. Dini's
theorem. The Weierstrass and Stone theorems. [Text: 7.14-15, 7.11-13,
7.26-33 (Skip proofs in 7.31-33 for the time being)]
2/2/15: The space of bounded functions. Two types of convergence in
it: uniform and pointwise. Completeness of this space. [Text: Chapter 7
through 7.10]
1/30/15: Homework is due at the beginning of the Friday
class. Discussion of homework. [Text: Chapter 6 through Theorem 6.6,
and 6.8, 6.12(a-d), 6.20-22]
1/28/15: The Fundamental Theorem of Calculus, integration by parts.
Riemann sums. Examples. [Text: Chapter 6: 6.21-22; Riemann sums are
not covered by the text, just by your class notes]
1/26/15: The proof of one of the properties of the integral. Geometric
interpretation. Differentiation of the integral. [Text: Chapter 6:
Theorems 6.12(a), 6.20]
1/23/15: Integrability of continuous functions, algebraic properties
of the integral. [Text: Chapter 6: Theorems 6.6, 6.8, 6.12(a-d),
6.13(b)]
1/21/15: Introduction, syllabus. Riemann integration theory: lower and
upper Riemann sums, the lower and upper integrals, the Riemann
integral.
[
Syllabus. Text: Chapter 6 through Theorem 6.5]
Last modified: (2015-11-22 14:24:24 CST)