Universität Wien

250029 VO Analysis 3 (2025S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Registration/Deregistration

Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 05.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 06.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 13.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 19.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 20.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 26.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 27.03. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 02.04. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 03.04. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 09.04. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 10.04. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 30.04. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 07.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 08.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 14.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 15.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 21.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 22.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 28.05. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 04.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 05.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 12.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 18.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 25.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Thursday 26.06. 08:00 - 09:30 Hörsaal 4 Oskar-Morgenstern-Platz 1 Erdgeschoß

Information

Aims, contents and method of the course

I expect students to already have working knowledge with the concepts and results presented in the following sections of the book Rudin, Walter. Analysis. 5. Auflage, De Gruyter Oldenbourg, 2022.

Kapitel 2: Einführung in die Topologie
- Metrische Räume
- Kompakte Mengen
Kapitel 4: Stetigkeit
- Grenzwerte von Funktionen
- Stetige Funktionen
- Stetigkeit und Kompaktheit
Kapitel 9: Funktionen mehrerer Variablen
- Lineare Abbildungen
- Differentiation
- Das Kontraktionsprinzip
- Der Satz über Umkehrabbildungen
- Der Satz über implizite Funktionen
Kapitel 10: Integration von Differentialformen
- Die Substitutionsregel

Students of the University of Vienna can access the text online via the link https://ubdata.univie.ac.at/AC17052312. The exercises in the book are a great preparation for the class.

In addition, you should be familiar with the term «homeomorphism» (meaning bijective maps between topological spaces that are continuous and have a continuous inverse) and the term «diffeomorphism between open subsets of Euclidean space» (meaning bijections that are differentiable and have a differentiable inverse).

I plan to cover the following topics.

Brief review of prerequisites: topology of Euclidean space; directional, total, and partial derivatives; inverse function theorem; transformation formula; characterization of isometries of Euclidean space

Linear algebra: projection formula; alternating multilinear forms in Euclidean space; Cauchy-Binet formula

Analytical tools: partitions of unity subordinate to an open cover; differential forms on domains (wedge product, exterior derivative, pull-back, closed and exact forms); divergence theorem; possibly Sard’s theorem

Curves in Euclidean space: length and reparametrization of curves; special reparametrizations of regular curves; geometry of plane curves (tangent field, normal field, curvature, total curvature, fundamental theorem of plane curves, osculating circles, angle functions, winding number and rotation number of periodic curves; homotopy invariance); possibly Cauchy’s theorem for holomorphic functions

Submanifolds of Euclidean space: submanifolds with and without boundary; tangent space; derivative of functions on submanifolds in direction of tangent vectors; integration of functions along submanifolds; orientability; integration of differential forms; Stokes’ theorem

Geometry of immersions into Euclidean space: Einstein summation convention; principles of map-mapping and the first fundamental form of immersions; tangential and normal projection; second fundamental form of immersions; Gauss equation; immersions of co-dimension one (Gauss map, principal curvatures, Theorema Egregium); reparametrization (geometric invariance of area, first and second fundamental form, curvature; local graph parametrization; uniform local graph parametrization in co-dimension one); possibly the geodesic equation

Differential topology: proof that closed differential forms on star-shaped domains are exact (Poincaré lemma); Brouwer’s fixed point theorem; hedgehog theorem; possibly the Jordan curve theorem

I will provide terse lecture notes for all topics except the review of prerequisites. Please attend the lecture course for additional details required, e.g., in the exam.

Assessment and permitted materials

Your grade will be based on a written 90-minute exam.
You are only allowed to bring writing utensils to the exam.

Minimum requirements and assessment criteria

You need to achieve at least half the marks on the final exam to pass this course.

If your percentage mark on the exam is p, then you will be awarded the following grade:

Sehr gut (1) for p in [87.5; 100]
Gut (2) for p in [75; 87.5)
Befriedigend (3) for p in [62.5; 75)
Genügend (4) for p in [50; 62.5)
Nicht genügend (5) for p in [0; 50)

Examination topics

All content discussed in class is examinable, unless explicitly declared otherwise.
In the exam, you will also be asked to apply the theory from class to problems of a similar complexity as those on the weekly problems sets.

Reading list


Association in the course directory

AN3

Last modified: We 26.03.2025 11:47