Overview of the course CALCULUS III (MATH 0240), Summer 2016, University of Pittsburgh
This Summer I shall be teaching CALCULUS III (MATH 0240) in the mathematics department at University of Pittsburgh. Mainly, we shall cover vectors, vector functions and space functions, arc length and curvature, functions of several variables, double and triple integrals, line integrals, surface integrals and some other related topics.
About the course
This is the third sequence of three calculus courses for science and engineering students. The goal is to prepare you to make use of calculus as a practical problem-solving tool.
Prerequisite
Math 0230 or equivalent, with grade of C or better.
Lectures and Recitations
- Lectures are from Monday to Thursday, 7-8.45 PM in 704 Thackeray Hall.
- Recitations are from Monday to Thursday, 6-6.50 PM in 704 Thackeray Hall. The recitation instructor for this class is Fawwaz Battayneh. His office hours are Monday and Thursday, 12-1 PM in the MAC.
- Moreover, you can also check the regular teaching page for this class: https://lupucezar.wordpress.com/teaching/ (Calculus 3-Summer 2016).
- Quizzes and exams will appear on the teaching section above.
Office Hours
My office hours are from Monday to Thursday, 5-6 PM in the MAC. If you want to schedule an additional meeting with me in my office (Thackeray hall, room 415) do not hesitate to send me an e-mail to lupucezar@gmail.com.
References (Lecture notes, textbook, and practice exams)
- P. Hajlasz, Lecture Notes in Calculus III, Part 1, Part 2, Part 3.
- J. Stewart, Essential Calculus, Early Transcendentals, 2nd edition.
- A. Athanas, Calculus 3-practice exams.
- I. Sysoeva, Calculus 3-practice exams.
- E. Trofimov, Calculus3-webpage.
- A. Yarosh, Calculus 3-webpage.
Grades
Your final grade will be determined as follows:
- Homework: 20% (2 homeworks of 10% each)
- Quizzes: 20% (4 quizzes of 5% each)
- Two exams: 60% (2 exams of 30% each)
- LON CAPA homework: 5% (extra credit)
Homework
I shall assign two homeworks which will appear on the teaching section. The first homework is here. Each one will be due before the exams. Additionally, one can do problems from the LON CAPA system. If you find it really annoying, you can also do this.
Final Grade Policy
A: 90-100%, A-: 85-90%, B+: 80-85%, B: 75-80%, B-: 70-75%, C+: 65-70%, C: 60-65%.
Exam Dates
The first exam will be on Thursday, July 14th, 7-9 PM in Thackeray 704.
The second exam will be on Wednesday, August 3rd, 7-9 PM in Thackeray 704.
Disability Resource Services
If you have any disability for which you are or may be requesting an accomodation, you are encouraged to contact both your instructor and the Office of Disability and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.
The 2016 Mathematical Olympiad Summer Program, June 7-July 2, Carnegie Mellon University, Pittsburgh, PA
The 2016 Mathematical Olympiad Summer Program (MOP or MOSP) is organized for the second time by the Mathematical Association of America (MAA) at Carnegie Mellon University (CMU) in the city of Pittsburgh between June 7th and July 2nd. The camp is run by the CMU faculty Po-Shen Loh (director) with the help of its associate director Razvan Gelca of Texas Tech University. They will be accompanied by other instructors (faculty, postdocs and Ph.D. students) from schools such as Massachusetts Institute of Technology (MIT), Stanford University, Harvard University, Columbia University, Carnegie Mellon University (CMU), North Carolina State University and University of Pittsburgh. This year’s MOP staff is here. The news about this year’s Mathematical Olympiad Summer Program appears on CMU’s front page here. This year’s guest speaker is Steve Shreve from CMU. Moreover, Noam Elkies from Harvard University will be the invited speaker for one of the evening’s seminars. Carnegie Mellon University also released a video about MOP:
Some details about the schedule are below:
- Classrooms are Gates 4101, Gates 4102, Scaife 125, and Wean 8220. The room numbers are indicated on the top row of the MOP schedule, in the leftmost column.
- Students will be separated into four groups.
- Black: USAMO winners and IMO team
- Blue: next top few from USAMO
- Red: students in grades 9 and 10.
- Many Red students will come with no prior MOP experience. Black level is quite impressive
- The timetable will be:
- 8:30am – 10:00am (Lecture 1)
- 10:15am – 11:45am (Lecture 2)
- 1:15pm – 2:45pm (Lecture 3), or 1:15pm – 5:45pm
- 7:30pm: optional-attendance evening research seminar
Moreover, this year is for the first time when MOP welcomes international students from countries such as Australia, Canada, Hungary, India, Romania, and Singapore. The two amazing Romanian students are Ioana Teodorescu and Stefan Tudose from the International Computer High School of Bucharest (ICHB). Another important news is the fact that four Romanians will serve as academic instructors. Besides the associate director, Razvan Gelca and myself, Bogdan Ion and Irina-Roxana Popescu of University of Pittsburgh are in the staff as well. This is for the second time in the history of MOP when this happens. The last time when MOP had four Romanian instructors was back in 2002. This was covered by Bogdan Suceava in the Romanian newspaper Evenimentul Zilei here. Also, the entire summary of courses is here. I shall deliver 15 lectures/ problem sessions & 1 seminar as follows:
1. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.
Group: black
Time and date: 8.30-10 AM, Thursday June 9th
Location: Wean Hall, Room 8820
SEMINAR. Arcsine power series expansion and Euler’s .
Groups: black. green, red 1 & 2
Time and date: 7.30-8.20 PM, Sunday June 12
Location: Stever Dorm
2. Real analysis techniques in solving elementary problems.
Group: black
Time and Date: 8.30-10 AM, Tuesday, June 14th
Location: Wean Hall, Room 8820
3. Real analysis techniques in solving elementary problems.
Group: blue
Time and Date: 1.15-2.45 PM, Tuesday, June 14th
Location: Scaife Hall, Room 125
4. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.
Group: red 2
Time and Date: 8.30-10 AM. Wednesday, June 15th
Location: Gates Hall, Room 4102
5. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.
Group: red 1
Time and Date: 10.15-11.45 AM, Wednesday June 15th
Location: Gates Hall, Room 4101
6. Fourier series and applications to elementary problems.
Group: blue
Time and Date: 8.30-10 AM, Thursday, June 16th
Location: Scaife Hall, Room 125
7. Fourier series and applications to elementary problems.
Group: black
Time and Date: 1.15-2.45 PM, Thursday, June 16th
Location: Wean Hall, Room 8220
8. Sequences, series of real numbers and inequalities.
Group: blue
Time and Date: 8.30-10 AM, Tuesday, June 21st
Location: Scaife Hall, Room 125
9. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.
Group: blue
Time and Date: 8.30-10 AM, Wednesday, June 22nd
Location: Scaife Hall, Room 125
10. Problems in computational geometry.
Group: red 2
Time and Date: 8.30-10 AM, Thursday, June 23rd
Location: Gates Hall, Room 4102
11. Problems in computational geometry.
Group: red 1
Time and Date: 10.15-11.45 AM, Thursday, June 23rd
Location: Gates Hall, Room 4101
12. Algebraic integers and applications.
Group: blue
Time and Date: 1.15-2.45 PM, Tuesday, June 28th
Location: Scaife Hall, Room 125
13. Algebraic integers and applications.
Group: red 2
Time and Date: 8.30-10 AM, Wednesday, June 29th
Location: Gates Hall, Room 4102
14. Algebraic integers and applications.
Group: red 1
Time and Date: 10.15-11.45 AM, Wednesday, June 29th
Location: Gates Hall, Room 4101
15. Irreducible polynomials in one variable and applications.
Group: blue
Time and Date: 8.30-10 AM, Friday, July 1st
Location: Scaife Hall, Room 125
PHILOSOPHY. Beyond MOP.
Groups: black, green, red 1 & 2
Time and date: 1.15-2.45 PM, Friday July 1st
Location: Stever Dorm
The William Lowell Putnam competition 2015 results (University of Pittsburgh)
The results of the 2015 William Lowell Putnam Mathematical Competition came out! The winner is Massachusetts Institute of Technology (MIT) followed by Carnegie Mellon University (CMU) and Princeton University in third place. There were 4275 contestants from over 500 universities in USA and Canada.
The University of Pittsburgh official Team (Stefan Ivanovici, Mingzhi Tian and Matthew Smylie) ranked 24th nationally and two of its members, Stefan and Ming, made it to top 460. In fact, Stefan made it to the top 200. Also, Derek Orr delivered a fine performance and he managed to make it to top 460 as well. Moreover, this news appeared on department’s front page.
This is the best performance as a team since 2002. Congratulations to all Pitt participants! It has been a great pleasure for George Sparling and I to work with such a group of talented students here at Pitt and this huge accomplishment is the result of two years of training. On the other hand, we want to mention and thank other faculty and graduate students involved in this year’s Putnam seminar: Gregory Constantine, Bogdan Ion, Kiumars Kaveh and Irina-Roxana Popescu. Moreover, we would like to acknowledge Bogdan’s involvement in previous year’s Putnam seminar. Some of these performances are listed below.
Pitt Team rank:
Initial value problems, arcsin power series expansion and Euler’s zeta(2)
In this post we shall discuss an ODE with initial conditions to derive a power series expansion for near . Let us consider the following initial value problem
.
It is quite clear that the function , satisfies the ODE above and since the coefficients are variable it is natural to look for a power series solution . Differentiation term by term yields
and .
Since , , our initial value problem will be equivalent to
.
Clearly, , , and for , . We easily obtain that for ,
and .
Therefore, this implies
.
The above formula serves as a good ingredient in evaluating series involving the central binomial coefficient like
This series converges at by Raabe’s test, and for it is uniformly convergent by the Weierstrass M-test.
Now, substitute , , and we get
.
Integrating from to , we have
.
On the other hand, Wallis’ formula (integral form) tells us that , and thus we finally obtain Euler’s celebrated
.
Remark. The ideas presented above have been generalized in this short paper.
Information on the Advanced Calculus II-MATH 1540 (undergraduate & graduate), Spring 2016
Advanced Calculus II-MATH 1540 (undergraduate version):
- There will be four homework assignments for this course assigned monthly. The problems will vary from standard to very difficult. I believe you are all already used to this approach after the first semester. In the end, I will help you with hints and solutions (for the difficult problems). It is important for you to learn and grasp the techniques used here. I am certain that they will be useful at some point in your life.
- Homework is 33% of your final grade. Dr. Sparling posted the syllabus here. Homework 1 has already been posted. See here.
- My office hours are Tuesday (4-6 PM) & Thursday (4-5 PM) in the MAC and by appointment in my office (Thackeray 415).
- Review sessions before the exams will be announced here and via e-mail or our Facebook group so stay tuned.
Advanced Calculus II-MATH 1540 (graduate version):
- Homework will be assigned by Dr. Xu and me and you can find it on my blog. I believe all the problems are suitable for the preliminary exam. Moreover, you should also have a look at the worksheets that will be posted on my blog.
- I will assign four HOMEWORK for this semester as follows:
- HW 1-LINEAR MAPS, LIMITS, CONTINUITY AND DIFFERENTIABILITY OF FUNCTIONS OF SEVERAL VARIABLES
- HW 2-THE INVERSE AND IMPLICIT FUNCTION THEOREMS; EXTREMUM PROBLEMS AND LAGRANGE MULTIPLIERS
- HW 3-INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES
- HW4-SETS OF MEASURE ZERO AND LEBESGUE INTEGRATION
- I encourage you to solve as many problems as you can from the homework. All of them have the same caliber as prelim problems from previous years.
- I would also suggest to keep an eye on the homework and exams from the undergraduate class.
- My office hours are Tuesday (4-6 PM) & Thursday (4-5 PM) in the MAC and by appointment in my office (Thackeray 415).
- Review sessions before the exams will be announced here and via e-mail or on our Facebook group so stay tuned.
Wallis sequence, Stirling’s approximation formula and some applications
- THE WALLIS PRODUCT FORMULA
In 1655, John Wallis wrote down the following celebrated formula:
In other words, we have the following nice representation for :
.
This amazing product formula, derived by Wallis by a method of successive interpolation, sparked attention for centuries for many mathematicians and it provides a key ingredient in the proof of the celebrated Stirling’s approximation formula. This formula will be discussed later in this post. Moreover, in 2015 researchers from University of Rochester found an unexpected connection between Wallis’ formula and quantum mechanics. In fact, it is shown that Wallis’ formula can be derived from a variational computation of the spectrum of the hydrogen atom. For more details about this discovery, one can see the paper here. Earlier, in 2007, Wastlund published a completely elementary proof here. One quick way to find it is by using Euler’s sine representation as a infinite product:
.
Taking , one has
,
and thus,
.
William Lowell Putnam Mathematical Competition at the University of Pittsburgh, 2015
The 76th annual William Lowell Putnam Mathematical Competition took place today, December 5th in 703 Thackeray Hall. Eleven Pitt students had the mission of tearing down the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Stefan Ivanovici, Matthew Smylie and Ming Tian. Other participating students were: Alec Jasen, Derek Orr, Tommy Bednar, Alex Mang, Jacob Gross, Andrew Tindall, Jack Hafer, and Mark Paulson. Below one can find this year’s Putnam problems. Congratulations to all participants!
Problem A1. Let and be points on the same branch of the hyperbola . Suppose that is a point lying between and on this hyperbola, such that the area of the triangle is as large as possible. Show that the region bounded by the hyperbola and the chord has the same area as the region bounded by the hyperbola and the chord .
Problem A2. Let and , and for . Find an odd prime factor of .
Problem A3. Compute
.
Problem A4. For each number , let
where is the set of positive integers for which is even. What is the largest real number such that for all ?
(As usual, denotes the greatest integer less or equal to .)
Problem A5. Let an odd positive integer, and let denote the number of integers such that and . Show that is odd if and only if is of the form with a positive integer and a prime congruent to or modulo .
Problem A6. Let be a positive integer. Suppose that are matrices with real entries such that , and such that and have the same characteristic polynomial. Prove that for every matrix with real entries.
Problem B1. Let be a three times differentiable function (defined on and real-valued) such that has at least five distinct real zeros. Prove that has at least two distinct real zeros.
Problem B2. Given a list of the positive integers , take the first three numbers and their sum and cross all four numbers off the list. Repeat with the three smallest remaining numbers and their sum . Continue this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: . Prove or disprove that there is some number in this sequence whose base representation ends with .
Problem B3. Let be the set of real matrices
whose entries (in that order) form an arithmetic progression. Find all matrices in for which there is some integer such that is also in .
Problem B4. Let be the set of all triples of positive integers for which there exist triangles with side lengths . Express
as a rational number in lowest terms.
Problem B5. Let be the number of permutations of such that
implies
for all . Show that for , the quantity
does not depend on , and find its value.
Problem B6. For each positive integer , let be the number of odd divisors of in the interval . Evaluate
.