250029 VO Analysis 3 (2025S)

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.