Analysis II for Physics, Uni Leipzig, SS 2020

This is a direct follow up of the WS 2019/20 Analysis & Linear Algebra for Physics lecture.

General rules to be admitted to exam, you will need to have at least 50% of the HW points (points for problems marked with (*) are extra points). In this semester HWs will not affect the final exam grade. HW points available at:

Due to pandemia, we start our semester in e-learning mode.

General outline: there will be

  1. Lecture notes with some level of interactivity
  2. Homeworks (HWs) posted online, to be solved and submitted according to the guidelines
  3. Tutorials by TA Mrs. Mahsa Sayyary Namin aimed at giving you opportunity to prepare for solving HWs and to understand better the material. More details at Mrs. Sayyary webpage:

Main topics:

  1. linear algebra, part II (six weeks)

-basics on tensors, -orthonormal tensors, -relation between tensors and matrices -decompositions: polar, spectral, -eigenpairs, -characteristic polynomial, -applications of decompositions, -Cayley-Hamilton formula

2. analysis, part II (nine weeks)

-Riemann integral, -basics of differentiation of functions of several variables, -inverse function theorem, -implicit function theorem, -introduction to ordinary differential equations


linear algebra references:

webpage of the 2019 course by Vitalii Konarovskyi:

Lankam, Nachtergaele, Schilling Linear Algebra, available at

Strang Introduction to Linear Algebra,

for vector/tensor calculus: part I of Gurtin, Fried, Anand The Mechanics and Thermodynamics of Continua

analysis references:

Rudin Principles of Mathematical Analysis (available at the 2019 course website)

Schueler Calculus 1-4,

Remark for students repeating Analysis II, who passed Analysis I not to my lecture (in previous years): the initial part on Linear Algebra in this semester is not mandatory for you. You have two options: either you choose to take it, which means you will have the exam identical with the rest of the group, or you choose not to take it, which means you will sit an alternative exam, with purely analysis questions (number and difficulty of questions in both cases will be the same).

Detailed plan:

week 1 (6-10 April)

  • HW9 (deadline shifted by +1 week compared to the one on the pdf) related to the part of the Linear Algebra section lectured in the previous semester: (the subproblem with (*) is not mandatory, i.e. will produce extra points if solved) attention: due to lack of tutorials on 10.04, I have extended the deadline considerably (it may coincide with the deadline for HW10, please take it into account when managing your time)
  • lecture notes (relevant part from the previous semester): they are rather dirty and should serve only to fill in gaps in your own notes from the lecture
  • no virtual lecture (theory already covered in the previous semester)
  • no tutorials: Good Friday

week 2 (13-17 April)

  • no HW
  • lecture notes: (we will have a Q&A session next week for your comments/questions)
  • tutorials Friday 17.04

week 3 (20-24 April)

week 4 (27 April-1 May)

week 5 (4 May- 8 May)

week 6 (11 May- 15 May)

  • no HW. current HW points available at:
  • lecture notes: The portion from week 5 covers also this week
  • no QA session: until Wednesday 13.05 morning it seems you have no questions: I received no emails.
  • tutorials Friday 15.05

week 7 (18 May- 22 May)

week 8 (25 May- 29 May)

week 9 (1 Jun- 5 Jun)

week 10 (8 Jun- 12 Jun)

week 11 (15 Jun- 19 Jun)

few additional matters:

partially due to my fault (inconsistent date for QA session in week 10 on webpage, sorry for that, no such problem in the email via Almaweb) the QA session of week 10 on Thursday 11.06 was not very crowded

there were some requests for more details on marked HWs. I suggest the following solution. You can find here: and here: more detailed distribution of points. Compare the solutions you have sent with the benchmark solutions on TA’s webpage. In case you absolutely disagree with your mark after making this comparison, please write to asking for feedback. (remark: links are newly created on 15.06, in case you had difficulties accessing the content before, try now again)

Exam details: 3.08.2020, 9:30-12. There will be five problems to be solved, ca 2 from linear algebra, 2 from Riemann integration, 1 from the remaining material. Every problem will be similar to either HW assignments or simpler proofs in the lecture notes. To be admitted to the exam you need at least 50% of your HW points. HW points do not influence exam grade. In case you have passed math1 not as my course, you are eligible for an alternative exam: 2 questions from linear algebra will be replaced by another 2 problems of analysis. To sit this alternative version, a student needs to send me until 17.07 his decision and proof of having math1 passed not to my course.

week 12 (22 Jun- 26 Jun)

week 13 (29 Jun- 3 Jul)

week 14 (6 Jun- 10 Jul)

  • HW15 (last one): A request from the marking person (repeated): use plane paper sheets for HWs! Take good quality scans! The marking person is entitled to reduce points if these conditions are not met, and in extreme cases give null points
  • lecture notes for week 14&15:
  • Figures to the lecture notes:
  • no QA session
  • tutorials Friday 10.07
  • detailed results of HW13

there are two ways to earn extra points for the exam admission: (i) each point from HW15 is doubled (details in the HW file), (ii) you can write a short lecture note (details in the lecture notes)

in week 15 a mock exam will appear, there will be QA session on Wednesday 15.07, and last tutorials. between 15.07 and 27.07 I am not available. on 28.07 the final point count will be made and thus the admission decision. in the time-frame 28.07-31.07 there will be an additional QA session.

exam presence: unfortunately, the faculty insists on the presence exam as the only option. those who cannot come will be given opportunity to take the resit exam as the first exam. the resit is (very roughly) expected in early October

week 15 (13 Jun- 17 Jul)

  • preparatory problems/mock exam
  • lecture notes for week 14&15 published last week
  • QA session Wednesday 15.07, 5pm at
  • tutorials Friday 17.07
  • results of HW14
  • please observe I am unavailable between end of our QA session and 27.07. so consider carefully whether you have any questions and ask them before or at QA session
  • in relation to question from QA session: you may have at the exam one A4 sheet with your notes

exam admission points announcement: the tentative HW points sum will appear on 28.07 in the morning. at noon I will have a short QA session concerning your possible complaints, after which the final grade will be given

the tentative admissions are here:

at 12:00 I will be available to discuss your complaints (eg concerning the analysis only case, it seems to me more of you asked for this than relevant emails I have found during the points count). you can contact me also via email before 22:00, when the final decision will appear

exam QA session: Wednesday 29.07, 12:00 CEST, at

exam admission: the list: will be send after 22:00 on 28.07, in case no complaints are made. then, after consulting with the Dean’s office, it becomes the official admission list

exam QA session note: It contains answer to questions raised, and a comment on Problem 9 (ii) (if a similar problem appears at the exam, it will be simpler). The bibliography to answer to question 2 is:

Lichnerowicz, Elements of Tensor Calculus

Gurtin, Fried, Anand The Mechanics and Thermodynamics of Continua

Dullemond, Peeters, Introduction to Tensor Calculus (a semi-translated version of the script can be found at

Schouten, Tensor Analysis for Physicists

exam admission announcement There are some discrepancies between the almaweb and my webpage admission list, I am currently consulting this with the Dean’s office. These discrepancies are:
concerning BIMA list (the first-timers): two more persons are admitted according to the almaweb
concerning BIPMA list (the repeaters): I missed to include the first person on the BIPMA list: 3752371 (sorry for this), and the system bans the following persons: 3745037, 3745851, 3708897 from the exam.
As soon as I know more, I will let you know.

exam admission final announcement: every person admitted on the BIMA2 part of the (old) list is admitted. concerning BIPMA2 part of that old list, there are the following changes: 3752371 is admitted, 3708897 can sit the exam, but needs to discuss their registration with the Dean’s office (the person failed to correctly register in almaweb), 3745851 and 3745037 are not admitted.

practical matters concerning the exam on Monday 3.08.2020:

please sparsely gather on the square by the entrance between chemistry and physics buildings at 9:45, wearing masks

Frau Schulthoff will let you into the building and guide you to Großer Hörsaal (205), you will enter from above, observe one-way walkways and 1,5m distance!

take places in 205 according to the ‘green scheme’, i.e. according to the green dots, see p.2 of

wear masks always when moving (entering, exiting, going to toilet) and when asking question. I recommend wearing masks all the time, but it is not obligatory when seated.

the exam starts at 10 am and ends at 12

after finishing, leave the 205 by the lower doors and exit the building following the arrows on the corridor floor

good luck!

tentative exam results: are in the column ‘grade’ of
The partial results are in the respective columns named 1-oa2
The passing threshold was 25/50, then affine dependence.

The solutions of exam problems and marking guidelines can be found at

In case you would like to discuss your exam in person, I will be available on Friday, 7.08 15-16 at A317, Neues Augusteum. Please write me an email beforehand, I need to let you into the building (side entrance close to the mdr tower).

retake exam: 25.09.2020. This retake will be available for

  • math2 BIMA students who were admitted (unlike usual semesters, also admitted students who did not show up at the first sitting are welcome)
  • math2 BIPMA list
  • math1, eligible for retake according to the normal rules (those who were admitted to the math1 exam and participated in it, but failed)
  • mathe1 (in German), group of Professor Hans-Peter Gittel

Exam format:

The difficulty of questions in both cases will be comparable to the respective first-sittings. To keep the overall exam comparable (now 4 problems to solve for math1 and math2), the sitting time for math1, math2 is reduced to 100 minutes. For mathe1 the number of problems is as it was, thus also the sitting time (120 minutes)

Due to covid disruption, I want to offer another retake, which will involve again math1 and math2. I am consulting this matter with Studienburo now, final decision soon.

practical matters concerning the exam on Friday 25.09.2020:

please sparsely gather on the square by the entrance between chemistry and physics buildings before 9 am, wearing masks

You will be let into the building and to Großer Hörsaal (205), observe one-way walkways and 1,5m distance!

take places in 205 according to the ‘orange scheme’, i.e. according to the orange dots, see p.4/5 of

wear masks always when moving (entering, exiting, going to toilet) and when asking question. I recommend wearing masks all the time, but it is not obligatory when seated.

the exam for math1 and math2 starts at 920 am and ends at 11

the exam for mathe1 (German group, lecture of Professor Gittel) starts at 920 am and ends at 1120am

after finishing, leave the 205 by the lower doors and exit the building following the arrows on the corridor floor

good luck!

solutions and marking remarks to retake exams:


math2:, problem 5 was part 2 of Thm 3.1 p. 3

tentative results:



in case of complaints, you will be able to view your solutions sheets and discuss your grade on 12.10.2020, between 11 am and 12 am in my office, A317 in Neues Augusteum. please read carefully the solutions before complaining

retake of both math1 and math2 will take place in the exam session of the winter semester 2020/21, most probably in February

math3 orientation meetingat 12 am on 12.10.2020 we will have a short orientation meeting on math3 course. let us meet in the ground floor of Neues Augusteum building (‘Foyer’) close to the entrance to the Gallery