Lectures
Summer Term 2009
Vektoranalysis
Prof. Freeden
Fr, 8:15 - 9:45, 46-220 (Beginn: 24.04.2009, 12 Termine)
Regelmäßige Termine für Übungen (14-tägig):
Nähere Informationen zu den Übungen
Anzahl der SWS: 2 Std. + 1 Std., (5 ECTS)
Unterrichtssprache:
Inhalt:
Die Vorlesung führt ein in die Integration von skalaren und vektoriellen Funktionen über Kurven und Flächen. Grundlegende Sätze, wie die Green'schen Formeln und die Sätze von Gauss und Stokes werden gezeigt. Inhalte der Vorlesung sind
- Parametrisierung von Kurven und (skalare und vektorielle) Kurvenintegrale
- Parametrisierung von Flächen und (skalare und vektorielle) Oberglächenintegrale
- Mannigfaltigkeiten und Tangentialräume, Differentiale differenzierbarer Abbildungen
- klassische Differentialoperatoren: div, grad, rot, Laplace
- Green'sche Formeln, Satz von Gauss, Satz von Stokes
- Anwendungen in der Physik
Literatur
- K. Jänich, Vektoranalysis. Zweite Aufl., Springer-Verlag, 1993.
- D.E. Bourne und P.C Kendall, Vektoranalysis. Teubner Studienbücher, 1973.
- F.E. Marsden und A.J. Tromba, Vektoranalysis. Spektrum Verlag, 1995.
Zur Vorlesung wird ein Skript erstellt, das am Ende der Vorlesungszeit für die Studenten zugänglich ist.
Voraussetzungen
Vorlesungen des ersten Studienjahres.
Leistungsnachweis und Prüfungen
Übungsschein bei aktiver Teilnahme an den Übungen und bestandener Klausur.
Geomathematics
Prof. Freeden
Di, 10:00 - 11:30, 48-582 (Beginn: 21.04.2009, 13 Termine)
Do, 15:30 - 17:00, 48-582 (Beginn: 23.04.2008, 11 Termine)
Regelmäßige Termine für Übungen:
Mi, 10:00 - 11:30, 48-438 (Beginn: 06.05.2009, 11 Termine)
Anzahl der SWS: 4 Std. + 2 Std., (4 ECTS)
Unterrichtssprache:
Content:
Geoscientific research influences new industrial developments as well as our everyday life. The possible reversion of the Earth's magnetic polarity in the next centuries and the indications of climatic changes are examples for open problems motivating further investigations. Nowadays, every aspect of the Earth's properties requires a sophisticated mathematical model. Moreover, present and future satellite missions represent in particular new mathematical challenges to scientists. As a consequence, modern mathematical methods are needed for the improvement of the Earth models and allow the development of advanced methods for solving the ocurring problems. This is the aim of geomathematics.
The lecture gives an introduction into selected topics of geomathematics. It discusses problems, like mass transport and mass distribution in the system Earth, the development of climate and weather models, the determination of the Earth's magnetic and gravity field as well as tools for solving the corresponding equation systems (by combined use of terrestrial and satellite data).
Literatur
- W. Freeden, T. Gervens, M. Schreiner: Constructive Approximation on the Sphere - With Applications to Geomathematics, Oxford Science Publication;
- W. Freeden: Multiscale Modelling of Spaceborne Geodata, Teubner;
- W. Freeden, V. Michel: Multiscale Potential Theory - With Applications to Geoscience, Birkhäuser;
- W. Freeden, M. Schreiner: Spherical Functions in Mathematical Geosciences (A Scalar, Vectorial, and Tensorial Setup), Springer;
- V. Michel: A Multiscale Method for the Gravimetry Problem - Theor. and Num. Aspects of Harmonic and Anharmonic Modelling, Shaker;
Voraussetzungen
Analysis and Linear Algebra
Leistungsnachweis und Prüfungen
'Uebungsschein' for successful participation in the tutorial; 4 credits can be obtained in an oral examination after the lecture.
Folgeveranstaltungen
This lecture is a good preparation for further lectures in the field of Geomathematics.
Sonstiges
The lecture will be given in English.
Spezielle Funktionen der Mathematischen (Geo-)Physik
Prof. Freeden
Di, 13:45 - 15:15, 48-538 (Beginn: 21.04.2009, 13 Termine)
Do, 10:00 - 11:30, 48-538 (Beginn: 23.04.2009, 11 Termine)
Regelmäßige Termine für Übungen:
Fr, 8:15 - 9:45, 48-438 (Beginn: 08.05.2008, 11 Termine)
Anzahl der SWS: 4 Std. + 2 Std., (4 ECTS)
Unterrichtssprache:
Content:
The lecture gives an elementary approach to the theory of special functions in mathematical physics with special emphasis on geophysically relevant aspects. The essential topics of the lecture are in chronological order: the Gamma function, orthogonal polynomials, spherical polynomials (scalar, vectorial, and tensorial case), and Bessel functions. All fields will be assisted by geophysically relevant applications.
Literatur
- W. Freeden, T. Gervens, M. Schreiner, Constructive Approximation on the Sphere (With Applications to Geomathematics), Oxford Science Publications, Clarendon, 1998.
- W. Freeden, V. Michel, Multiscale Potential Theory (with Applications to Geoscience), Birkhäuser Verlag, Boston, 2004.
- N.N. Lebedev, Spezielle Funktionen und ihre Anwendungen, Bibliographisches Institut, Zürich, 1973.
- C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer, 1966.
- C. Müller, Analysis of Spherical Symmetrics in Euclidean Spaces, Springer, 1998.
- W. Freeden, M. Schreiner: SphericaFr, 08:15 - 09:45 l Functions in Mathematical Geosciences (A Scalar, Vectorial, and Tensorial Setup), Springer;
Voraussetzungen
Analysis and Lineare Algebra
Leistungsnachweis und Prüfungen
'Uebungsschein' for successful participation in the tutorial; 4 credits can be obtained in an oral examination after the lecture.
Folgeveranstaltungen
The lecture is a good preparation for further activities in the field of geomathematics such as "Constructive Approximation", "Potential Theory", "Inverse Problems", etc.
Inverse Problems
Dr. Martin Gutting
Mo, 10:00 - 11:30, 44-456 (Beginn: 20.04.2009, 13 Termine)
Mi, 8:15 - 9:45, 49-538 (Beginn: 23.04.2009, 12 Termine)
Regelmäßige Termine für Übungen:
Mi, 13:45 - 15:15, 49-506 (Beginn: 22.04.2008, 13 Termine)
Anzahl der SWS: 4 Std. + 2 Std., (4 ECTS)
Unterrichtssprache:
Content:
Inverse problems today appear in many technological problems. If one wants to know the reason of a measured effect, one has to cope with an inverse problem. For example in computer tomography, the attenuation of x-rays is measured after they have passed the object of interest (e.g. the human body). The cause for the attenuation is the density of the object. From the mathematical point of view, inverse problems consist of inverting certain operator equations of the first kind. In the equation Ax=y, y is given and x is wanted. An inverse problem is especially concerned with the case that y is not given exactly, y is not in the image of A or A^-1 is not continuous, which is the case if A is compact and D(A) is not finite dimensional. Inverse problems have to be regularized (stabilized), to avoid the appearing amplification of errors. The lecture will give a mathematical introduction for the solution and the regularization of inverse problems with concrete geomathematical and other technological applications.
Literatur
- Louis, A.: Inverse und schlecht gestellte Probleme. Teubner 1989.
- Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer 1992.
- Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer 1996.
- Rieder, A.: Keine Probleme mit inversen Problemen. Vieweg 2003
Voraussetzungen
Vordiplom
Seminar zur Geomathematik (Seminar)
Prof. Freeden
Inhalt:
Some of the presentations of the seminar can be found
here.
Vorbesprechung:
wird noch bekanntgegeben
Termin:
Blockveranstaltung am Ende des Semesters, Termin wird noch bekanntgegeben.
Prof. Dr. Willi Freeden,
Dr. Carsten Mayer,
HDoz. Dr. V. Michel,
Dr. habil. M. Schreiner
Anzahl der SWS: 2 Std., (0 ECTS), n. V..
Inhalt:
Diplomanden- und Doktorandenseminar (ergänzt durch Vorträge renommierter Gäste). Diskussion forschungsrelevanter Themen der Geomathematik.
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