Mathematik | Informatik
Nicola Casale, 2003 | Unterentfelden, AG
This thesis is meant to give students on the upper secondary school level an introduction to Gauss’s theorem. It is a very powerful tool to determine the flux through a closed surface more efficiently. It is only discussed in three-dimensional space.
Introduction
The research question that guided me throughout my work was: How can the flux of a vector field through a closed surface be calculated?
Methods
The aim of this thesis is to explain the statement of Gauss’s theorem and to demonstrate, how we can derive Newton’s law of universal gravitation from an infinitesimal version of this law by using Gauss’s theorem. This was done in the so-called case study. In addition, a simple unidimensional mathematical proof was given.
Results
The most important result of this work is to get to know Gauss’s theorem and to learn how powerful it is. In the case study, it was observed that the gravitational field of a sphere behaves inversely proportional to the square of the distance from the centre. The whole mass seems to be concentrated in the centre of the sphere.
Discussion
The mathematical concepts presented may not be explained as precisely as in literature. These inaccuracies may occur in definitions and explanations due to the high level of mathematical understanding required. Another shortcoming is that the concepts were only defined in three-dimensional space. In fact, they are defined for n dimensions. Furthermore, although it was explained how the surface and volume elements must be replaced when using polar or cylindrical coordinates, the derivation of the Jacobian matrix was not explained. On the other hand, illustrations were used where appropriate to facilitate understanding, and, where calculations have been carried out, the process has been explained step by step.
Conclusions
Gauss’s theorem is a very powerful tool for calculating the flux through a closed surface, because it not only makes the calculation simpler but also allows to find the flux faster. How a surface is parametrised is crucial, because different kinds of volume elements are needed, depending on which coordinate system is used. The way in which a surface is parametrised could be explored in much greater depth and improve the general understanding of the topic.
Würdigung durch den Experten
Dr. Fabian Ziltener
In dieser Arbeit fasst Nicola Casale mit viel Einsatz und Elan den Divergenzsatz von Gauß, ein klassisches Resultat der Analysis, zusammen. Er erklärt, wie man mit Hilfe dieses Satzes das Newtonsche Gravitationsgesetz herleiten kann. Er beweist auch den eindimensionalen Fall des Satzes. Mit seiner Arbeit macht Herr Casale dieses fortgeschrittene Thema Lesern und Leserinnen zugänglich, die über eine mathematische Ausbildung auf dem Niveau der Sekundarstufe II verfügen. Die Arbeit ist ansprechend gestaltet und enthält viele farbige Abbildungen, welche die Konzepte gut verdeutlichen.
Prädikat:
gut
Neue Kantonsschule Aarau
Lehrer: Dr. Maik Berchtold