MATH 448-ANALYSIS II- SPRING 2018-A. FREIRE

1/11 Th  Lecture postponed (due to JMM 2018 San Diego)

1/16  Tu  Riemann integral in 1 vble: Darboux definition. Intrgrability of: monotone fns, cont. fns,
fns w/ D_f finite. Sets of measure zero. Lebesgue's characterization of Riemann integrability (statement).

1/18 Th Lebesgue's integrability theorem (proof). Convergence of  integrals: examples, theorem on uniform convergence.
HW set 1 (due 1/25)

1/23 Tu Fundamental theorem of Calculus. Cantor set, Lebesgue's function. Lebesgue outer measure.

1/25 Th The problem of measure. Vitali's covering theorem (statement). Dini's derivate numbers.
Lebesgue's theorem on differentiability a.e. of monotone functions (outline of proof.)
Lebesgue's theorem on differentiability a.e. of monotone functions
(includes three problems= HW2, due Feb. 1)

1/30 Tu Functions of bounded variation: basic properties
Functions of bounded variation

2/1 Th Functions of bounded variation (continued)
HW set 3 (due 2/8): exercises 4, 5, 7 and 8 from the handout "functions of bounded variation"

2/6 Tu Rectifiable curves

2/8 Th Absolute continuity

2/13  Tu Riemann-Stieltjes integrals
Problems on Riemann-Stieltjes integrals
(HW 4, due 2/20)

2/15  Th Differential forms and line integrals
Riemann-Stieltjes integrals and line integrals of one-forms (15 pages)
(Includes exercises 1 to 5=HW 5, due Tu 2/27)

2/20 Tu Exterior differential, closed forms, exact forms.

2/22 Th Integration in several variables: continuous functions

2/27 Tu Sard's theorem (equidimensional), Lebesgue's integrability thm: proofs.

3/1 Th: MIDTERM 1. Included: material up to 2/22

Midterm 1 (with solutions)

3/6 Tu Integration over more general sets--Jordan content, J-measurable sets
Jordan Content and Riemann integral

3/8 Th  L-Measurable sets in R^n ([Fleming], 5.1 and 5.2)
HW 6: (due 3/22): Problems 1 and 2 from handout "Jordan content"; problems 1, 3, 12 from [Fleming], p.180/181.

3/13, 3/15: SPRING BREAK

3/20: measurable sets (cont'd)/integral of functions with cpt support (5.10, 5.3)

3/22: integral of bounded measurable functions over bounded measurable sets (5.4)

3/27: measurable functions: monotone approx by simple functions (Fleming 5.10)/ Luzin's theorem

3/29: convergence theorems for the integral ([Fleming, 5.11])
HW 7: (due 4/5)--Fleming,  p.226: 3, 5/ Fleming p. 236: 4, 5, 6
Reading assignment: section 5.7 (Change of measure under affine transformations)--will be assumed.

4/3 Fubini's theorem
convergence theorems and Fubini's theorem

4/5 Tonelli's theorem

4/12 MIDTERM 2: lectures from 3/6 through 4/5
Midterm2 (with solutions)

4/17 Equidimensional area formula

4/19 Change of variables theorem/ Banach indicatrix/ area formula for non-injective maps
Area formula and change of variables
HW 8 (due Thursday 4/26): (5.7) 4, (5.8) 4, 5, 7, 8

4/24 Area and integration on surfaces

4/26 Gauss-Green theorem
Integration on manifolds and Gauss-Green theorem
Review problems: Fleming (8.4) 2,6/ Exercises 1, 2 in this handout

Review session: Tuesday May 1st, time 11AM

FINAL EXAM:  May 2nd (Wed.), 12:30--2:30 (comprehensive)
Final Exam (with solutions)