### CONTACT INFO

OFFICE 415 (4TH FLOOR), THACKERAY HALL
E-MAIL: lupucezar@gmail.com, cel47@pitt.edu

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## 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.

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)

1. P. Hajlasz, Lecture Notes in Calculus III, Part 1, Part 2, Part 3.
2. J. Stewart, Essential Calculus, Early Transcendentals, 2nd edition.
3. A. Athanas, Calculus 3-practice exams.
4. I. Sysoeva, Calculus 3-practice exams.
5. E. Trofimov, Calculus3-webpage.
6. A. Yarosh, Calculus 3-webpage.

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 hereEach 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.

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 integral formula in spherical coordinates system

Fubini’s theorem

Green’s theorem

## 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

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

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## 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:

• 2014-108th
• 2013-98th
• 2002-17th
• 2001-18th
• 2000-23rd
• 1998-13th (best ever performance!)
• 1997-15th

From left to right: Jonathan Rubin (undergraduate director), Stefan Ivanovici, yours truly, Matthew Smylie and Derek Orr

From left to right: Derek Orr, Matthew Smylie, yours truly, Stefan Ivanovici and George Sparling

From left to right: Derek Orr, Stefan Ivanovici, Matt Smylie, yours truly, Alex Mang, Jack Hafer, George Sparling, Tommy Bednar and Mark Poulson

## 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 $\arcsin^2(x)$ near $x=0$. Let us consider the following initial value problem

$\displaystyle (1-x^2)y^{{\prime}{\prime}}-xy^{\prime}-2=0, y(0)=y^{\prime}(0)=0$.

It is quite clear that the function $g:(-1, 1)\to\mathbb{R}$, $\displaystyle g(x)=\arcsin^2(x)$ satisfies the ODE above and since the coefficients are variable it is natural to look for a power series solution $\displaystyle y(x)=\sum_{n=0}^{\infty}a_{n}x^n$. Differentiation term by term yields

$\displaystyle y^{\prime}(x)=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^n$ and $\displaystyle y^{{\prime}{\prime}}(x)=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n$.

Since $a_{0}=y(0)=0$, $a_{1}=y^{\prime}(0)=0$, our initial value problem will be equivalent to

$\displaystyle 2a_{2}+6a_{3}x+\sum_{n=2}^{\infty}((n+2)(n+1)a_{n+2}-n^2a_{n})x^n=2$.

Clearly, $a_{2}=1$, $a_{3}=0$, and for $n\geq 2$, $\displaystyle a_{2n+1}=0, a_{n+2}=\frac{n^2}{(n+2)(n+1)}a_{n}$. We easily obtain that for $n\geq 2$,

$\displaystyle a_{2n+1}=0$ and $\displaystyle a_{2n}=\frac{(2^{n-1}(n-1)!)^2}{(2n)(2n-1)\ldots 4\cdot 3}=\frac{1}{2}\frac{2^{2n}}{n^2\binom{2n}{n}}$.

Therefore, this implies

$y(x)=\boxed{\displaystyle \arcsin^2(x)=\frac{1}{2}\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2\binom{2n}{n}}}, |x|\leq 1$.

The above formula serves as a good ingredient in evaluating series involving the central binomial coefficient like

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2\binom{2n}{n}}=\frac{\pi^2}{18}, \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3\binom{2n}{n}}=\frac{2}{5}\zeta(3), \sum_{n=1}^{\infty}\frac{1}{n^4\binom{2n}{n}}=\frac{17}{3456}\pi^4.$

This series converges at $x=1$ by Raabe’s test, and for $x\in (-1, 1)$ it is uniformly convergent by the Weierstrass M-test.

Now, substitute $x=\sin t$, $0, and we get

$\displaystyle t^2=\sum_{n=1}^{\infty}\frac{2^{2n-1}}{n^2\binom{2n}{n}}\sin^{2n}t$.

Integrating from $0$ to $\frac{\pi}{2}$, we have

$\displaystyle \frac{\pi^3}{24}=\sum_{n=1}^{\infty}\frac{2^{2n-1}}{n^2\binom{2n}{n}}\int_0^{\frac{\pi}{2}}\sin^{2n}t$.

On the other hand, Wallis’ formula (integral form) tells us that $\displaystyle\int_0^{\frac{\pi}{2}}\sin^{2n}tdt=\frac{\pi}{2^{2n+1}}\binom{2n}{n}$, and thus we finally obtain Euler’s celebrated

$\displaystyle\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$.

Remark. The ideas presented above have been generalized in this short paper.

## Information on the Advanced Calculus II-MATH 1540 (undergraduate & graduate), Spring 2016

• 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.

•   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:
•  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

1. THE WALLIS PRODUCT FORMULA

In 1655, John Wallis wrote down the following celebrated formula:

$\displaystyle\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\ldots=\lim_{n\to\infty}\prod_{k=1}^{n}\frac{4k^2}{4k^2-1}=\lim_{n\to\infty}\prod_{k=1}^n\left(\frac{2k}{2k-1}\cdot\frac{2k}{2k+1}\right)=\frac{\pi}{2}.$

In other words, we have the following nice representation for $\pi$:

$\displaystyle\pi=\lim_{n\to\infty}\frac{1}{n}\left(\frac{2\cdot 4\ldots (2n)}{1\cdot 3\ldots (2n-1)}\right)^2=\lim_{n\to\infty}\frac{1}{n}\left(\frac{(2n)!!}{(2n-1)!!}\right)^2$.

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:

$\displaystyle\boxed{\sin x=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{\pi^2n^2}\right)}$.

Taking $x=\frac{\pi}{2}$, one has

$\displaystyle 1=\frac{\pi}{2}\prod_{n=1}^{\infty}\left(1-\frac{1}{(2n)^2}\right)=\frac{\pi}{2}\prod_{n=1}^{\infty}\left(\frac{(2n)^2-1}{(2n)^2}\right)$,

and thus,

$\displaystyle\frac{\pi}{2}=\prod_{n=1}^{\infty}\left(\frac{(2n)^2}{(2n-1)(2n+1)}\right)=\prod_{n=1}^{\infty}\frac{4n^2}{4n^2-1}$.

## 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!

SESSION A:

Problem A1. Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1$. Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB$.

Problem A2. Let $a_{0}=1$ and $a_{1}=2$, and $a_{n}=4a_{n-1}-a_{n-2}$ for $n\geq 2$. Find an odd prime factor of $a_{2015}$.

Problem A3. Compute

$\displaystyle\log_{2}\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$.

Problem A4. For each number $x$, let

$\displaystyle f(x)=\sum_{n\in S_{x}}\frac{1}{2^n},$

where $S_{x}$ is the set of positive integers $n$ for which $[nx]$ is even. What is the largest real number $L$ such that $f(x)\geq L$ for all $x\in (0,1)$?

(As usual, $[z]$ denotes the greatest integer less or equal to $z$.)

Problem A5. Let $q$ an odd positive integer, and let $N_{q}$ denote the number of integers $a$ such that $0 and $gdc(a, q)=1$. Show that $N_{q}$ is odd if and only if $q$ is of the form $p^{k}$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.

Problem A6. Let $n$ be a positive integer. Suppose that $A, B$ are $n\times n$ matrices with real entries such that $AM=MB$, and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\displaystyle\det (A-MX)=\det (B-XM)$ for every $n\times n$ matrix $X$ with real entries.

SESSION B:

Problem B1. Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $\displaystyle f+6f^{\prime}+12f^{{\prime}{\prime}}+8f^{{\prime}{\prime}{\prime}}$ has at least two distinct real zeros.

Problem B2. Given a list of the positive integers $1, 2, 4, \ldots$, take the first three numbers $1,2, 3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4, 5, 7$ and their sum $16$. Continue this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6, 16, 27, 36, \ldots$. Prove or disprove that there is some number in this sequence whose base $10$ representation ends with $2015$.

Problem B3. Let $S$ be the set of $2\times 2$ real matrices

$\displaystyle M= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

whose entries $a, b, c, d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$.

Problem B4. Let $T$ be the set of all triples $(a, b, c)$ of positive integers for which there exist triangles with side lengths $a, b, c$. Express

$\displaystyle\sum_{(a, b, c)\in T}\frac{2^a}{3^b5^c}$

as a rational number in lowest terms.

Problem B5. Let $P_{n}$ be the number of permutations $\pi$ of $\{1, 2, \ldots, n\}$ such that

$|i-j|=1$ implies $|\pi (i)-\pi (j)|\leq 2$

for all $i, j\in \{1, 2, \ldots, n\}$. Show that for $n\geq 2$, the quantity

$\displaystyle P_{n+5}-P_{n+4}-P_{n+3}+P_{n}$

does not depend on $n$, and find its value.

Problem B6. For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate

$\displaystyle\sum_{k=1}^{\infty} (-1)^{k-1}\frac{A(k)}{k}$.

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