in-person lectures on Thursdays and Fridays 9:15-11, Theoretical Lecture Hall. some of these lectures are streamed via https://conf.fmi.uni-leipzig.de/b/jan-jvp-zhe-h5l
in-person tutorials on Fridays 13:30-15, Small Lecture Hall
office hours 15-16 on Fridays, room 225 (the same, physics building)
materials/literature I recommend taking notes at the lecture, I am not following a single source; having said that
for the first part of the course (multivariate integral)
- quite close to what we are doing is the script by Artem Saposhnikov, available at http://www.math.uni-leipzig.de/~konarovskyi/teaching/2019/Math3/pdf/Sapozhnikov_Math3_LectureNotes.pdf
- if you want to dig deeper, I recommend (parts of) the book by V. Zorich Mathematical Analysis II . or by Munkres Analysis on Manifolds
- there are plenty of interesting materials at the math3 course site by V. Konarovskyi, including further reading and some exercises http://www.math.uni-leipzig.de/~konarovskyi/teaching/2019/Math3/Math3_2019.html
- I will post the handwritten lecture notes via moodle, the lecture notes of the last year (that you should take with a pinch of salt) are available at https://drive.google.com/file/d/1gh4iWMPUfhdFqyAQjXcGNrVcrMXuf-NE/view?usp=sharing
overview of the semester:
Part A. Multivariate integration theory and vector calculus
- Recalling/updating integration theory from math1/2
- Riemann/Darboux definitions, their equivalence
- basic properties of integrals
- Peano-Jordan measure
- elementary sets
- inner/outer measure, its properties
- pathological sets
- definition of Peano-Jordan measure
- zero-measure sets
- set-theoretical properties of Peano-Jordan measure
- equivalence of measurability and boundary having zero measure
- graph of a continuous functions is zero measure
- Introduction to integration theory of multivariable, real functions
- partitions of measurable sets
- Riemann/Darboux definitions in multi-d
- basic properties of multi-d integrals (algebraic ones, set-theoretic ones, continuity on closure implies integrability)
- three important inequalities (Minkowski, Holder, Jensen)
- Iterated integrals and Fubini
- 2d Fubini: normal sets, proof of 2d Fubini, examples
- 3d and multi-d Fubini, examples
- ultranormal sets, remarks on relaxing assumptions in Fubini theorem
- Change of variables
- 1d motivation
- diffeomorphisms
- change of variables formula, sketch of proof
- examples: angular, cylindrical, spherical
- Improper integrals
- Elements of more advanced material: more general assumptions (in Fubini, variable change) thanks to Lebesgue theory, example of ‘failure’ of Fubini, monotone and dominated convergence theorems, differentiation of integral with a parameter and ‘Feynman’s trick’
- Introduction to integration of real functions on manifolds
- curves, tangent vector, regular and arc-length parametrisation, Frenet-Serret frame
- manifold: definition, examples, tangent and normal spaces, zero sets
- Integrations of vector fields on manifolds
- integrals along curves, conservative/ potential vector fields, curl-free fields, Green’s theorem
- integration over manifolds of codimension-1, Gauss’ theorem
- classical Stokes theorem
Part B. Ordinary Differential Equations
sources:
- lecture notes,
- oversimplified source: selected topics from two initial Chapters of Boyce&DiPrima Elementary Differential Equations and Boundary Value Problems, 11th edition
- best-match source: selected topics from Chapter 1 of Nagy Ordinary Differential Equations, notes homepage https://users.math.msu.edu/users/gnagy/teaching/ode.pdf (with some links to related solution videos)
- slightly more advanced reference: Teschl Ordinary Differential Equations and Dynamical Systems, book’s homepage: https://www.mat.univie.ac.at/~gerald/ftp/book-ode/
- notation, Initial Value Problem
- linear first order ODEs, examples
- Hadamard well posedness
- non-linear first order ODEs: examples of troublemakers, Bernoulli, exact and semi-exact ODEs
- general theory: Picard-Lindelof, its extensions, Peano, stability via Gronwall