Real Analysis II
Spring 2011
Contact Information Important Dates
Professor:Dr. Scott Nollet Exam 1Wednesday, February 23
Office:310 TTC
Office Hours: MTRF 1:30pm-4:30pm, or by appointment. ProjectWednesday, April 27
Office Phone:257-6339
Home Phone:920-7988
E-mail:s.nollet@tcu.edu
Class Hours: MW 4:30-5:50pm, 244 TTC Final ExamWednesday, May 4
3:00pm-5:30pm
Pre-requisite:Real Analysis I
Texts: The primary texts is Real Mathematical Analysis by Charles Pugh, but I will also use Understanding Analysis by Stephen Abbott (Springer-Verlag) and possibly also Advanced Calculus by Gerald Folland.
Goals: Having rigorously developed the foundations of calculus of one variable in Real Analysis I, the continuation explores more advanced topics. We'll start with uniform convergence of functions and power series in one variable, then move on to higher dimensional topics, including topology, derivatives and integration, including differential forms and Stoke's theorem, which provides a generalization of the fundamental theorem of calculus to higher dimensions. If time allows, I will discuss function spaces, which are infinite-dimensional.
Grades: Grades will be based on Homework (30%), two Exams (20% each) with tentative dates listed above, and the Final (30%). I will regularly assign homework and encourage you to discuss it with your classmates.

When all the coursework is complete, grades will be assigned on the following basis (though note that the grade D is not allowed for graduate students). Let P be the percentage of points scored. Then

P Grade
P at least 85 A
P at least 83, but less than 95 B/B+/A-/A, depending on strength of final
P at least 75, but less than 83 B
P at least 73, but less than 85 C/C+/B-/B, depending on strength of final
P at least 65, but less than 73 C
P at least 63, but less than 65 D/D+/C-/D, depending on stregth of final
P less than 63 D or F
Attendance and make-up policy: University policy states that regular and punctual class attendance is essential and that no assigned work is excused due to absence, no matter what the cause.
Exams and/or Projects: There will be two exams and the final, however it is possible that I will substitute a project for one of the exams.
Disabilities statement: Texas Christian University complies with the Americans with Disabilities Act and Section 504 of the Rehabilitation Act of 1973 regarding students with disabilities. Eligible students seeking accommodations should contact the Coordinator of Student Disabilities Services in the Center for Academic Services located in Sadler Hall, 11. Accommodations are not retroactive, therefore, students should contact the Coordinator as soon as possible in the term for which they are seeking accommodations. Further information can be obtained from the Center for Academic Services, TCU Box 297710, Fort Worth, TX 76129, or at (817) 257-7486.
Academic Misconduct (see section 3.4 from Student Handbook) Any act that violates the academic integrity of the institution is considered academic misconduct. The procedures used to resolve suspected acts of academic misconduct are available in the offices of Academic Deans and the Office of Campus Life. Specific examples and more details are in the Student Handbook.
Homework Assignments
HW #SectionProblemsDate Due
1 6.2-3 6.2 #1,3,4,6,8a-e, 6.3 #2,3 Wednesday, 1/19
2 6.4-5 6.4 #1,2,3,7, 6.5 #1,4,7,8ab Monday, 1/31
3 6.6 #1,7,8,A Wednesday, 2/9
4 Pugh, Ch. 1 # 24abd,26,44a Monday, 2/14
5 Pugh, Ch. 2 # B, 3,8,14a (b for extra credit),24 Wednesday, 2/23
6 Pugh, Ch. 2 # 40,54,55,71,77 Wednesday, 3/2
7 Pugh, Ch. 5 # 1,3,4a,c-e,13 3/30
8 Pugh, Ch. 5 # C, 20, 24, 32 4/11
9 # D,E,F
A. For the function g(x) on page 180 of Abbott's book, compute g'(0) and g''(0).
B. Show that the open ball of radius 1 centered at the origin in Rn is homemorphic to Rn itself.
C. Find the critical points of the function f(x,y)=xy(12-3x-4y). Compute the Hessian at each critical point. Find the eigenvalues to determine if the critical point is a max, min, or saddle point. Also write out the third order Taylor expansion at (1,1).
D. Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,z)=x2+4y-6z on the half ellipsoid x2+2y2+3z2=125 with z non-negative.
E. (a) Show that f(x) is continuous at c if and only if w(c) = 0.
(b) Show that for r > 0, the set Dr of x where w(x) is at least r is a closed set.
F. Use induction and integration to give a formula for the volume of the closed n-ball of radius r in Rn.