**lectures** on Mondays and Wednesdays 9:15-11 at

https://miserv3.mathematik.uni-leipzig.de/b/jan-gyh-fzx

all lectures will be recorded and made available

LaTeXed lecture notes will be prepared by a group of students. If anyone wants to join this group, please let me know. Depending on performance of the noting persons, they may be granted extra points needed for exam admission.

**virtual office hours** on Monday, at 1:30pm

**tutorials** Fridays TBA

**exam admission **upon homeworks, with possible extra assignments

**contact**: burczak@math.uni-leipzig.de**course outline: **Ordinary Differential Equations (4 weeks), vector calculus/Stokes theorem (ca 4 weeks), complex analysis/Fourier analysis (ca 5 weeks), Partial Differential Equations (ca 3 weeks

**Part I: Ordinary Differential Equations**

linear first order ODEs; Hadamard well-posedness; Bernoulli equation/logistic ODE as an example; exact equations/existence of potential and solvability via ImplicitFT, geometrical interpretation, separable, semi-exact ODEs; Picard-Lindeloef theorem, global/local versions, Picard iterates, Gronwall and stability, first order systems, Duhamel formula, 2×2 case in detail/applied to second order equations, autonomous systems, qualitative analysis of 2×2 case (phase portraits)

sources:

- lecture notes,
- selected topics from three initial Chapters of Boyce&DiPrima
*Elementary Differential Equations and Boundary Value Problems, 11th edition* - selected topics from two initial Chapters of Nagy
*Ordinary Differential Equations,*notes homepage https://users.math.msu.edu/users/gnagy/teaching/ode.pdf (with some links to related solution videos) - as a slightly more advanced reference: Teschl
*Ordinary Differential Equations and Dynamical Systems,*book’s homepage: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/

**week 1**: first order ODEs

lectures: Monday (26.10), Wednesday (28.10),

homework (HW1) on Wednesday 28.10 with deadline 9 days

tutorials Friday 1pm at https://bbb.linxx.net/b/mar-6aw-eke

the materials are available via BBB (recording of lectures, it seems it takes around a day after the recording for them to appear) and moodle (virtual blackboard notes, homework, and preliminary notes for myself for lectures)

**week 2**: first order ODEs, general theory

lectures, tutorials, homework analogously as before, see moodle. office hours 1:30 on math3 room of the bigbluebutton (same as lecture)

**week 3**: first order systems/second order equations

lectures, tutorials, analogously as before, see moodle. office hours 1:30 on math3 room of the bigbluebutton (same as lecture).

HW3 published on 12.11 with an extended due date: 23.11. There will be no HW in week 4. In week 4 there will be one lecture, with a short summary of the ODE submodule, and tutorials on Wednesday or on Friday, as you prefer (to be decided on 13.11 tutorials)

**week 4**: first order systems, autonomous systems, qualitative analysis of 2×2 case (phase portraits), summary of ODE material

in this week we sum up our ODE submodule. there is one lecture on Monday and office hours at 1:30 on math3 room of the bigbluebutton (same as lecture) and tutorials on Wednesday in the lecture time, at the tutorials bbb room. no HW and no tutorials on Friday

**Part II: Integration in R^n, vector calculus**

Peano-Jordan content, Riemann and Darboux definitions of Riemann integral for multivariate real functions, basic properties, Fubini theorem, variable change formula, Stokes and related theorems, vector calculus identities

reading, apart from the lectures:

- lecture notes of A. Sapozhnikov, can be found at http://www.math.uni-leipzig.de/~konarovskyi/teaching/2019/Math3/pdf/Sapozhnikov_Math3_LectureNotes.pdf
- math3 teaching materials of V. Konarovskyi, including his lecture notes (the link above)
- last two chapters of lecture notes of C. DeLellis (see moodle)

**week 5**: Peano Jordan content, its properties

lectures: Monday (23.11), Wednesday (25.11),

no homework

tutorials Friday https://bbb.linxx.net/b/mar-6aw-eke, hours as agreed with TA Marius Neubert

**week6**: Peano Jordan content and its properties continued, Riemann and Darboux definitions of Riemann integral for multivariate real functions and its basic properties,

lectures: Monday (30.11), **Friday **(4.12) (due to Dies Academicus on Wednesday, we will have lecture in the Friday tutorials slot)

HW4

**week7**: Fubini theorem and change of variables,

HW5

**week8 **change of variables continued, recalling math2 material on manifolds

the teaching week is shortened by the Rectorate to Mon-Wed. therefore, there will be tutorials in the Wed lecture slot, i.e. the week’s plan is:

lecture on 14.12, tutorials on 16.12 at 9:15

**offline week (**materials via moodle, prepared over the extended university lockdown, available on 8.01.2021)

integrals over manifolds: scalar functions, vector functions, needed elements of vector calculus

**week 10-11 **(week 9 within Christmas break)

classical Stokes-type theorems, summary of the second module

**Part III: Fourier Transform and basic PDEs**

**week 12**

Fourier Transform: definition, basic properties, Schwartz class, inversion formula, Riemann-Lebesgue lemma

**week 13 **

basic PDEs via Fourier Transform: heat equation (Gauss-Weietrstass kernel, some qualitative properties including infinite speed of propagation), Schrodinger equation (via heat), Poisson/Bessel equation, wave equation (the last shifted to tutorials).

summary of module 3

**Timeline before the exam**

You will have tutorials 2 or 3 more times (including that of 5.02), depending on your TA decision. He will communicate the relevant details.

The current HW8 is due on Wednesday 10.02.

On 15.02 I will publish a mock exam. Those who have troubles with being admitted to the exam (insufficient HW points) may send me solved mock problems. The deadline for this will be very short (ca 12 hours).

On 16.02 the final exam admission decision will be announced.

There will be office hours on 17.02 at 9:30 am, at the usual math3 bbb site.

The **exam** is planned for 9:15-11:45, 22.02, via moodle. Its problems will be similar to the mock problems.

The persons who want **math1 or math2 resit** should contact me via email.

math1 and math2 exams:

**math1 resit is on 15.02**, together with the group of Dr. Stephane Mescher. In case of you encounter large discrepancies between the material covered by me and by Dr. Mescher in math1, please let me know.

the next **math2 resit** will be after the** incoming spring semester** 2021 (i.e. probably late July), and it will be together with the math2 course exam that starts in April: so please keep in touch with that course for possible material discrepancies. No math2 exam this exam session.