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Gauss Theorema Egregium
2001 (Swedish)Independent thesis Advanced level (degree of Master (One Year))Student thesis
Abstract [en]

The essay is divided into two parts, one dealing with curves and the other one dealing with surfaces. The curve theory introduces a special class of curves, called regular curves. To a regular curve it also belongs a tangent vector field which assigns a tangent vector at each point of the curve. Other things to be treated are how to calculate the length of a curve, how to measure the bending(curvature) of a curve in space and how to know if a curve lies in a plane or not(one talks about torsion). The chapter ends with the Fundamental Theorem of Curves which shows that the curvature and torsion of a curve completely describes the curve, except for its position in space. In the theory of surfaces one also works with a certain class of them, called simple surfaces. Mathematically, a surface is defined be a collection of simple surfaces that overlaps. For simple surfaces one can define concepts like tangent vectors, normal vector, metric coefficients, Christoffel symbols, the first- and second fundamental form and so on. The last part of chapter 2 will be devoted to different kinds of curvatures, which helps us measure how a surface is bending. In order to accomplish that, one can study curves whose image lies on the surface. The so-called normal curvature of such a curve will measure how the surface bends along the curve, but the most effective tool to study how the surface is bending is the Weingarten map which is the direction derivative of the normal to the surface. The Weingarten map can be represented by a matrix whose eigenvalues are called principal curvatures, the determinant is called Gaussian curvature and half-the-trace is the mean curvature. Finally, this essay deals with a remarkable theorem in the theory of surfaces, Gauss Theorema Egregium. It says that the Gaussian curvature, which is defined in terms of the second fundamental form, in fact, only depends on the first fundamental form. Gauss was so surprised by the fact that he had to call the theorem ”egregium” (extra-ordinary). Finally, I would like to thank my tutor, Lektor Ilie Barza for his help and guidance, and my friends Måns Englund, Rex and Hermez for their support.

Place, publisher, year, edition, pages
2001. , 35 p.
Identifiers
URN: urn:nbn:se:kau:diva-54798Local ID: MAT D-1OAI: oai:DiVA.org:kau-54798DiVA: diva2:1104562
Subject / course
Mathematics
Available from: 2017-06-01 Created: 2017-06-01

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  • apa
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  • de-DE
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  • en-US
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  • nn-NB
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  • Other locale
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Output format
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