MATH 448- ADVANCED CALCULUS II- SPRING 2017- A. FREIRE

Course outline

Syllabus

COURSE LOG

W 1/11 Functions of bounded variation
Functions of bounded variation

F 1/13 BV: saltus + continuous decomposition of BV
HW 1: exercises 1 to 5, sections 1 and 2 of the notes. (Due 1/20)

M 1/16 MLK Holiday (no lecture)

W 1/18 precompactness in BV (Theorems of Helly and Fubini); rectifiable curves

F 1/20 Lebesgue outer measure on the line

M 1/23 Cantor sets of arbitrary measure/Lebesgue's singular function

W 1/25 Absolute continuity (start)

F 1/27 Absolute continuity/ measurable sets in R^n ([Fleming, sect. 5.2])
HW2 due (problems 6-10 in BV handout)

M 1/30 Measurable sets in R^n
HW3 (due Friday Feb 3.) Fleming p. 180: 1, 3, 6, 7, 12.

W 1/30 Lebesgue-mesurable sets/ algebras, sigma-algebras

F  2/3 Measurable sets, measurable functions

M 2/6 Measurable functions and covergence
HW4 (due Monday Feb 13): [Fleming] p. 226: 2, 3, 4
Also problems 1 and 2 in this handout:
Measurable Functions

W 2/8 Theorems of Egorov and Luzin (see handout)

F 2/10 Integration of bounded measurable functions (Fleming 5.3, 5.4)

M 2/13 Riemann integral and Jordan content
Jordan content and Riemann integral
(includes 3 problems= HW 5, due Monday 2/20) Also 4 questions for "extra credit" (w/ proof or counterexample)

W 2/15 Riemann integrability criteria (see handout)

F  2/17 Riemann integral vs. Lebesgue integral/ Leb. integrability of general functions/ monotone approx. by simple functions
(see [Fleming, 5.11]. Also, [Fleming, 5.6] is a reading assignment (won't be covered in class)).

M 2/20 Convergence theorems for the Lebesgue integral (Fleming 5.11)

W 2/22 Convergence theorems (cont)
HW6 [Fleming, 5.11] 4, 5, 6, 8, 9 10 (due Monday 2/27)

F 2/24 Fubini's Theorem
Convergence Theorems and Fubini's Theorem
(includes 4 problems.= HW 7, due W 3/8)

M 2/27 Fubini's Theorem (cont)

W 3/1 Area formula (intro, linear case)

F 3/3  Area Formula (linear case/ equidimensional Sard's theorem)

M 3/6 Area formula (conclusion)
Area Formula

W 3/8 Extensions of the area formula/ Banach indicatrix

F 3/10 Differentiation under integral sign (5.12)

M 3/13 to F 3/17: Spring Break

M 3/20: L^p spaces
Lp spaces-notes and problems (HW 8, due F 3/24)

W 3/22: Approximation in L^p spaces

F 3/24: approximation (cont.)

M 3/27: discussion of HW/ approx in L^p by smooth functions w/ compact support

W 3/29 MIDTERM (with solutions)

F 3/31: Vitali's covering theorem, Lebesgue points
Lebesgue differentiation, Vitali covering
(includes two problems)

W 4/5: Differential forms, line integrals, exact and closed forms
HW9: problems from the 3/31 handout and, from [Fleming]: 7(6.2), 7(6.3), 5(6.4)
(due Monday 4/10) HW9 solutions

F 4/7: locally exact forms, Poincare lemma, line integrals and homotopy
Differential forms and line integrals

M 4/10: calculus of differential forms/ 2-forms, exterior derivative, invariance under maps

W 4/12: invariance of line integrals/ integrals of 2-forms in R2/ Stokes' theorem.
HW 10- due Wed 4/19: four problems in the differential forms handout
HW 10 solutions

F 4/14: no lecture

M 4/17: measure and integral on manifolds (start)

W 4/19: measure on manifolds: invariance, partitions of unity, graphs

F 4/21: Gauss-Green theorem for C^1 domains (start)
Integration on manifolds and the Gauss-Green theorem

M 4/24: Gauss-Green thm for C^1 domains (end)

HW 11-due Friday 4/28 (three problems from [Fleming]) (8.3) 9 /(8.4) 2, 6
and both problems in the handout.

W 4/26: Divergence theorem/weak derivatives/Sobolev space (intro)

F 4/28: Remarks on weak derivatives and approximations (see handout, section 6)

FINAL EXAM (comprehensive): Friday, 5/5 8:00-10:00AM
Final Exam
Solutions