Time and place
Lecture: Mi, 8-10h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced
Content
This compulsory lecture for teacher training students in the M.Ed. builds on the basic lectures Analysis I and II and supplements them with the following two main topics:
\textit{Multidimensional integration:} The one-dimensional Riemann integral known from Analysis I is generalized to real-valued functions of several variables, for which a suitable instrument for measuring the content/volume of multidimensional sets is first introduced with the Jordan content. Then the classical integral theorems (transformation theorem, Fubini's theorem) are derived, and path and surface integrals are considered. With the help of the divergence and rotation of vector fields, the two aforementioned integral types can be related to each other using the integral theorems of Gauß and Stokes, which considerably simplifies the calculations in practical applications.
\textit{Complex Analysis:} In contrast to Analysis I, here the (complex) differentiability of functions of a complex variable is examined. As will be shown, complex differentiable, so-called holomorphic functions are subject to much stricter rules and laws than their real-valued counterparts, which leads to both beautiful and surprising results. To this end, we will prove Cauchy's intergal theorem and Cauchy's integral formula and take a closer look at applications and conclusions from these.
Previous
knowledge
Analysis~I and II, Linear Algebra~I and II
Usability
Further Chapters in Analysis