Jacob Shapiro's Teaching Page at the Math Department of Princeton U.

Analysis 1 Recitation Sessions

The course website contains logistical information as well as the homework and their solutions. Below are "scripts" (summaries) of the recitation sessions I held:

• Recitation Session of Week 1
• Colloqium of Week 2--The Peano Axioms (from Halmos).
• Recitation Session of Week 2
• Colloqium of Week 3--Dedekind Cuts (from Rudin).
• Recitation Session of Week 3
• Colloqium of Week 4--The Cantor Set (Rudin and Koenigsberger).
• Recitation Session of Week 4
• Colloqium of Week 5--Moebius Transformations: Interactive HTML5 Demo, Nice video demonstration and Mathematica Demo.
• Recitation Session of Week 5
• Colloqium of Week 6--Metric Spaces with S^2 and RP^2.
• Recitation Session of Week 6
• Colloqium of Week 7--Limit Points and Dense Sets.
• Recitation Session of Week 7
• Colloqium of Week 8--Space-Filling Curves.
• Recitation Session of Week 8
• Colloqium of Week 9--Continuity, Continuous Extensions, and the Pasting Lemma.
• Recitation Session of Week 9
• Colloqium of Week 10--Compactness.
• Recitation Session of Week 10
• Colloqium of Week 11--An example of a continuous nowhere differentiable real function (From Rudin PMA theorem 7.18 pp. 154). Nice visualization of this function in Mathematica.
• Recitation Session of Week 11
• Colloqium of Week 12--Smooth non-Analytic Functions.
• Recitation Session of Week 12
• Colloqium of Week 13--Review for Midterm.
• Recitation Session of Week 13 - Midterm.
• Colloqium of Week 14--Post-Midterm Review.
• Recitation Session of Week 14

If you plan to submit your homework, it would make me very happy if you used LyX.

Analysis 2 Recitation Sessions

The course website contains logistical information as well as the homework and their solutions. Below are "scripts" (summaries) of the recitation sessions I held:

• Recitation Session of Week 1
• Hints for Homework 1
• Solutions for Homework 1
• Hints for homework 2
• Solutions for Homework 2
• Recitation Session of Week 3 (updated with two more examples)--Frechet and Gateaux derivatives.
• A note about the Zeta Function and 1+2+3+4+.... For more details see this.
• Solutions for Homework 3
• Recitation Session of Week 4--Continuity of the Frechet derivative, derivative of power series in Banach algebra, differentiable but not continuously differentiable examples.
• Hints for homework 4
• Recitation Session of Week 5--Convex functions, multivariable optimization and multi-indexn notation.
• Hints for homework 6--Euler Lagrange Procedure.
• Recitation Session of Week 7--Inverse and Implicit Function Theorems, Diffeomorphisms, Submanifolds.
• Discussion of Homework 8--Linear Differential Equations in R^n, and "Flows".
• Hints for homework 9.
• Recitation Session of Week 10.
• Solutions for Homework 10--fixed the end of Q1.
• Solutions for Homework 11--Q1 till Q4.
• Recitation Session of Week 12--No lesson held, only hints.

If you plan to submit your homework, please consider typing it into the computer using, for example, .

Full hand-written notes (in German, 91MB) were kindly provided by Simon Vonlanthen.

Quantum Field Theory 2 Recitation Sessions

The course website contains logistical information as well as the homework and their solutions. Below is some relevant material:

• Solution for HW1.
  See also pages 4 till 13 about the harmonic oscillator in this document and pages 6 concerning a worked out example of a perturbed harmonic oscillator.
• Expansion around the classical solution to the equations of motion in scalar field theory, tree level perturbation theory, scalar QED.
• On week 11 (May 13th 2016) we discussed the BRST symmetry. For background see Peskin Chapter 16. See a calculation involving BRST symmetry which was started in class. See also a BRST Primer by Dennis Nemeschansky, which discussed BRST in other contexts (not just gauge theories).
• See the book of Nakahara for more information about the geometric formulation of gauge theory.
• Our final exercise involved section III and V of Coleman and Weinberg's classic paper. See this stackexchange discussion about how to verify equation (5.18) solves (5.15).

If you plan to submit your homework, please consider typing it into the computer using, for example, .

Analytical Mechanics Recitation Sessions

The course website contains logistical information as well as the homework and their solutions, the lecture notes, and forums in which you may ask questions.
I warmly recommend the book by V. I. Arnold which you may download electronically freely as ETHZ students. Another book to consider is H. Goldstein's.
Below is some relevant material from the sessions I held:
I usually will provide the unredacted solutions after the submission date has passed, for those people who are struggling with the German official solutions.

Week	Summary of Recitation Session	Redacted or Full Solutions
I	No classes
II	Organization info, some simple gradient exercises, and hints for HW1	Full Solutions of HW1
III	Common pitfalls for HW1 and hints for HW2	Full Solutions for HW2
IV	Common pitfalls for HW2 and hints for HW3	Full Solutions for HW3
V	Common pitfalls for HW3 and hints for HW4	Full Solutions for HW4
VI	Common pitfalls for HW4	Full Solutions for HW5
VII	a discussion about natural units and the difference between central and conservative forces	Full Solutions for HW6
VIII	No summary of the recitation session this week. Most of the presentation which was done in class is contained in the (redacted) solutions.	Full Solutions for HW7 and a Mathematica notebook for the plots.
IX	Some remarks about natural frequencies	Full Solutions for HW8
X	No summary for this week	Full Solutions for HW9
XI	No summary for this week	Full Solutions for HW10
XII	A short note about Legendre Transformations A note about Hamilton's Least Action Principle seen as zero of the Frechet derivative of the action	Full Solutions for HW11
XIII	Flows	Full Solutions for HW12
XIV	The Hamilton-Jacobi equation and Poincare Transformations	Full Solutions for HW13

If you plan to submit your homework, please consider typing it into the computer using, for example, .

Mechanics of Continua Recitation Sessions

The course website contains logistical information as well as the homework and their solutions, the lecture notes, and forums in which you may ask questions.
Below is some relevant material from the sessions I held:
I usually will provide the unredacted solutions after the submission date has passed, for those people who are struggling with the German official solutions.

Week	Summary of Recitation Session	Redacted or Full Solutions
I	Some Reminders from Analysis 2 and Linear Algebra	Redacted Solutions of HW1
II	Differential Geometry Perspective on Curvilinear Coordinates, Official Sol-n of HW1Q2	Redacted Solutions of HW2
III	None	Redacted Solutions of HW3
IV	None	Redacted Solutions of HW4
V	None	Redacted Solutions of HW5
VI	There will be a note about the finite element method to solve the Navier equation.	Not yet
VII	None	Hints for HW7

Additional Material:

• Complex Line Integrals
• The Existence and Uniqueness of the Neumann Problem on Bounded Domains

If you plan to submit your homework, please consider typing it into the computer using, for example, .

General Relativity Recitation Sessions

The course website contains logistical information as well as the homework and their solutions, the lecture notes, and forums in which you may ask (public) questions.

As for textbooks (in order of relevance to you):

• Robert Wald's General Relativity book. Apparently if you Google it the first result is a scan of the book. This book is not chatty but is very precise. Note that the popular Sean Carroll book heavily borrows from Wald, and is generally loquacious in an excessive way.
• Norbert Straumann's General Relativity book, which is freely available over SpringerLink if you're on the ETH intranet. Not my favorite introduction but a great source to find explicit analysis of many problems which remain absent in many other books, for example the post Newtonian approximation or the Kerr-Newman solution.
• One must also mention Misner, Thorne, Wheeler which is overwhelmingly encyclopedic, Steven Weinberg who attempted to obscure differential geometry from his description, but still contains many useful examples.
• As far as math books go, my favorites for differential geometry are Boothby's and Bredon's (very concise introduction). Also see Barrett O'Neill's book on semi-Riemannian geometry. There are also geometry books especially geared towards physics applications: Nakahara and Frankel.
  
  Below is some relevant material from the sessions I held:
  I usually will provide the unredacted solutions after the submission date has passed.

  Please note that this material is provided as is with no guarantees for correctness. For success in the exam the official solutions must be consulted as the ultimate source.

  Week	Summary of Recitation Session	Redacted or Full Solutions
  I	Top. Mnflds., Diff. Mnflds., examples and counter-examples, tensor product of vector spaces and a few words on vector bundles. Note that in class I forgot to mention the basic fact that an open subset of a smooth manifold is a smooth manifold again. Together with the fact det is continuous and the complement of the singleton zero is open, this is a better way to see why the general linear group is a smooth manifold, and that its dimension is n^2.	Full
  II	The Basis of the Tangent Space induced by a Chart, Properties of the Transition Matrices, The Flow Generated by a Vector Field, The Pushforward and the Pullback, and the Lie Derivative	Full
  III	Parallel Transport and the second tangent bundle--a very sketchy summary	Full
  IV	Affine connections, R-linear vs. F(M)-linear, affine spaces, Christoffel symbols via connection, Levi-Civita connection corresponding to a metric(doesn't exist yet)	Full
  V	no summary	Full
  VI	no summary	Redacted
  VII	Integration	Redacted
  VIII	No summary	Full
  IX	No summary	Full
  X	No summary	Full
  XI	No summary	Redacted
  XII
  XIII
  XIV

  If you plan to submit your homework, please consider typing it into the computer using, for example, .

Proseminar in Disordered Transport

During the spring semester of 2018 I had the pleasure to mentor three graduate students in Oded Zilberberg's Proseminar on transport in disordered media. What my group concentrated on mainly dealt with using super-Gaussian integrals (like path integrals) a la Efetov's 1995 manuscript to re-write the DC conductivity of a disordered material, and after a stationary phase approximation, derive a non-linear sigma model. This model can then be analyzed, and ultimately one can prove Poisson statistics in a localized regime, or non-zero conductivity can be shown in the delocalized regime. The students gave oral and written reports, the latter of which are available below:

• Anton Eder - The DC conductivity via Kubo, review of path integrals, and introduction to super-Gaussian integrals.
• Jeremy Mann - The Saddle-Point Approximation and arrival at a NLSM.
• Tim Fleischmann - Calculation of the first quantum corrections to the Drude formula using the NLSM formalism.

Calculus I Lecture -- Math UN1101 (Section 02)

Columbia's fine print:

• The faculty statement on academic integrity.
• The Columbia College Honor Code.
• The Columbia University Undergraduate Guide to Academic Integrity.
• Students are expected to do their own work on all tests and assignments for this class and act in accordance with the Faculty Statement on Academic Integrity and Honor Code established by the students of Columbia College and the School of General Studies. Because any academic integrity violation undermines our intellectual community, students found to have cheated, plagiarized, or committed any other act of academic dishonesty can expect a conversation and may be referred to the Dean’s Discipline process. It is students’ responsibility to ensure their work maintains the standards expected and should you have any questions or concerns regarding your work, you can: (a) Talk with your TA (b) Ask the instructor (c) Refer to the Columbia University Undergraduate Guide to Academic Integrity.

General information:

Location:	207 Mathematics Building
Time:	Mondays and Wednesdays 4:10pm-5:25pm (runs through January 23rd until May 6th 2019 for 28 sessions)
Recitation Sessions	Fridays 2pm-3pm in Hamilton 602 run by Donghan Kim. First session on Feb 1st, 2019. Attendance not mandatory but warmly recommended.
Office hours:	Tuesdays 6pm-8pm and Wednesdays 5:30pm-7:30pm (or by appointment) in 626 Mathematics (starting Jan 29th)
Teaching Assistants:	Donghan Kim dk2571@columbia.edu Mat Hillman mh3691@columbia.edu Ziad Saade zs2351@columbia.edu
TAs office hours:	Will be held in the help room (502 Milstein Center) at the following times: – Mat: Fridays 10am-12pm. – Donghan: Thursdays 4pm-6pm. – Ziad: Tuesdays 4pm-6pm.
Misc. information:	Calculus @ Columbia
Getting help:	Your best bet is the office-hours, and then the help room. If you prefer impersonal communication, you may use the Piazza website to pose questions (even anonymously, if you're worried). TAs will monitor this forum regularly.
Textbook:	Lecture notes are available and will be updated on a weekly basis. I will do my best to follow the the Columbia Calculus curriculum so as to make sure you can go on to Calculus II smoothly. I will use material from various textbooks, some of which include: Spivak, Apostol, Courant and sometimes Stewart just to set the timeline (since this is what Columbia's curriculum is based upon).
Homework assignments:	Assignments will appear on this website after before the Monday lecture every week and are due at the the Monday lecture of the following week. You do not have to hand in assignments every week, but you may present your solutions to the TAs or me to gain extra credit (see below) You must hand-in your assignments weekly to a satisfactory level in order to be able to take the exam. If there are reasons why you cannot hand-in your assignment at some week please write me an email directly.
Grading:	There will be two midterms during lecture time (see below for dates) and one final (after the last lecture). Your grade is (automatically) the higher of the following two options: [Option A] The final carries 50% weight, the midterms each carry 25% weight for a total of 100% of your grade. [Option B] The final carries 40% weight, the midterms each carry 25% weight for a total of 90% of your grade. The remainder 10% is given to you if you succeed in accomplishing the following task at least three times during the semester: show up (no appointment necessary) to one of the office hours of the TAs or me, successfully orally present a solution to one of the homework assignments whose solution has not been published yet (should take you not more than 10 minutes max), and in the same occasion submit a neatly written down solution to the thing you just presented. Due to the results of the first midterm I have decided to change the rules a bit. You are now required to hand-in homework assignments weekly. If you do to a satisfactory level you may come to the exam and will get 10 points of your final grade. I am still undecided about how to divide the grades between the final and the second midterm, but you will not be negatively affected in any way if you scored low on the first midterm.

Tentative Schedule (will be updated to reflect what happened in class as the weeks elapse):

Date	Week	Lecture #	Contents	Special events
Wed Jan 23	I	1	Sets	First class
Mon Jan 28	II	2	Sets and functions	HW1 published (note typos corrected!); HW1 solutions published
Wed Jan 30	II	3	Functions
Mon Feb 4	III	4	Limits of sequences	HW2 published ; HW2 solutions published
Wed Feb 6	III	5	Limits of sequences
Mon Feb 11	IV	6	Limits of functions	HW3 published ; HW3 solutions published
Wed Feb 13	IV	7	Limits of functions and continuity	Practice Midterm 1 published Practice Midterm 1 Detailed Solutions published (note: After careful review and getting some feedback, I have decided that the actual midterm will be of the same format, but contain less questions on each part, and also that partial credit will be given to your calculation (contrary to the instructions in the practice midterm as it stands now) in case your answer is incorrect. So your explanation could help you salvage points if your final answer is wrong. As before, the material for the midterm is everything in the lecture notes from the beginning up to and not including derivatives).
Mon Feb 18	V	8	Review
Wed Feb 20	V	9	Midterm I	Midterm 1 and Midterm 1 solutions
Mon Feb 25	VI	10	Derivatives 3	HW4 published ; HW4 solutions published
Wed Feb 27	VI	11	Derivatives 4
Mon Mar 4	VII	12	Derivatives 5	HW5 published (typo fixed) HW5 Solutions published
Wed Mar 6	VII	13	Derivatives 6
Mon Mar 11	VIII	14	Derivatives 7	HW6 published ; HW6 Solutions published
Wed Mar 13	VIII	15	Derivatives 8
Mon Mar 25	IX	16	Review
Wed Mar 27	IX	17	Midterm 1.5	Practice Midterm 1.5 published ; Practice Midterm 1.5 Solutions published ; HW7 published HW7 Solutions published ; Midterm 1.5 published ; Midterm 1.5 Solutions published ;
Mon Apr 1	X	18	Integrals 1	HW8 published ; HW8 Solutions published
Wed Apr 3	X	19	Integrals 2
Mon Apr 8	XI	20	Integrals 3	HW9 published ; HW9 Solutions published
Wed Apr 10	XI	21	Integration: Fundamental theorem of calculus
Mon Apr 15	XII	22	Applications of derivatives: extremal points	HW10 published ; HW10 Solutions published ; Practice Midterm 2 published ; Practice Midterm 2 Solutions published
Wed Apr 17	XII	23	Applications of derivatives: Convexity
Mon Apr 22	XIII	24	Review of practice midterm 2
Wed Apr 24	XIII	25	Midterm 2	Midterm 2 published ; Midterm 2 Solutions published
Mon Apr 29	XIV	26	Review towards final	HW11 published ; HW11 Solutions published
Wed May 1	XIV	27	Review towards final
Mon May 6	XV	28	Review towards final	Practice Final published ; Practice Final Solutions published ; HW12 published ; HW12 Solutions published
Mon May 13 4:10pm	XV	*		Final published and Final Solutions published .

PHY103 - Newtonian mechanics

No material has appeared here--all content was collected on Princeton's canvas website.

PHY104 - Electricity and magnetism

• Week 1 - Jan 26,28 and 31 2022.

MAT201 - Multivariate Calculus

MAT330 - Complex Analysis with Applications

• Lecture Notes (PDF).
• Assignments:

  Assignment	Due date	Automatic extension	Solutions	Notes
  HW1v3 (typo in Q4 corrected; numbering fixed)	Feb 10th 2023	Feb 12th 2023	HW1 Solutions
  HW2v4 (domain in Q7 fixed, Q6 rephrased)	Feb 17th 2023	Feb 19th 2023	HW2 Solutions
  HW3v2	Feb 24th 2023	Feb 26th 2023	HW3 Solutions
  HW4v2	Mar 3rd 2023	March 5th 2023	HW4 Solutions
  Practice Midterm			Practice Midterm Solutions	Don't worry if Q2 was hard.
  Midterm			Midterm Solutions
  HW5v3	Mar 31st 2023	April 7th 2023	HW5 Solutions	Some hints have been added, and importantly, Q5 and Q12 are now [extra] credit; in v3 the contour for Q18 is fixed.
  HW6v3	April 7th 2023	April 9th 2023	HW6 Solutions	Q9&Q10 revised; v3: Q11 fixed index of composition.
  HW7v4	Apr 14th 2023	April 16th 2023	HW7 Solutions	v4 fixed the phrasing of the uncertainty principle and the conformal questions.
  HW8v3	Apr 21st 2023	April 23rd 2023	HW8 Solutions	Phrasing in Q9 fixed again (v3).
  HW9	Apr 28th 2023	April 30th 2023	HW9 Solutions
  HW10v2	Not to be submitted		HW10 Solutions	A few more exercises have been moved to extra credit in v2.
  Practice final	Not to be submitted		Practice final solutions
  Final	May 14th 2023		Final solutions

MAT 520: Functional Analysis

• Lecture notes. See also Jaksic lecture notes.
• HW1 due Sep 15th 2023; HW1 Solutions.
• HW2 due Sep 22nd 2023; HW2 Solutions.
• HW3 due Sep 29th 2023; HW3 Solutions.
• HW4 due Oct 6th 2023; HW4 Solutions.
• HW5 due Oct 13th 2023; HW5 Solutions.
• Midterm; Midterm Solutions.
• HW6 due Oct 27th 2023; HW6 Solutions.
• HW7 due Nov 3rd 2023.
• HW8 due Nov 10th 2023.
• HW9 due Nov 17th 2023.
• HW10 due Dec 1st 2023; HW10 Solutions.
• HW11 due Dec 8th 2023; HW11 Solutions.
• HW12 not to be submitted.
• Final; Final Solutions.

MAT 595 / PHY 508: Topics in Mathematical Physics: Mathematical Aspects of Condensed Matter Physics

Lecture notes.

MAT330: Complex Analysis

Lecture notes.

MAT520: Functional Analysis

Lecture notes.

HW1.

HW2.

HW3.

HW4.

HW5.

HW6.

HW7.

HW8.

HW9.

HW10.

Midterm.

Final.

Final Solution.

MAT425: Measure Theory

Lecture notes.

HW1, HW1 Solutions.

HW2, HW2 Solutions.

HW3, HW3 Solutions.

HW4, HW4 Solutions.

HW5, HW5 Solutions.

Midterm, Midterm solutions.

HW6, HW6 Solutions.

HW7, HW7 Solutions.