| Code |
15619
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| Year |
1
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| Semester |
S2
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| ECTS Credits |
8
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Essential knowledge of Real Analysis and Linear Algebra.
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Learning outcomes |
It is intended that the student knows:
1. Parameterize regular curves in plane and space and determine their curvature and torsion;
2. Classify curves based on their curvature and torsion;
3. Recognize and parameterize regular surfaces in space and determine their curvatures using maps;
4. Determine the geodesics of a regular surface;
5. Classify some special surfaces;
6. Study of minimal surfaces as critical points of the area function.
7. Extend the concepts learned for surfaces in space to abstract manifolds of dimension 2.
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Syllabus |
Chapter 1 Curves in the Plane: Parameterized curve; Reparameterization by arc length; Signed curvature. Frenet-Serret equations; Examples; Fundamental Theorem
Chapter 2 Curves in Space: Examples; Curvature and torsion; Trihedron and eq. from Frenet-Serret; Fundamental Theorem
Chapter 3 Surfaces: Surfaces. Examples; Gauss map; Applications on surface and tangent vectors; Orientable surface; First Fundamental Form; Weingarten map; Normal curvature, average curvature and Gaussian curvature; Second Fundamental Form; Ruled surfaces and surfaces of revolution; Minimal surface; Gauss's Egregium Theorem; Geodesic and normal curvatures of a curve; Geodesics; Umbilical points; Mainardi-Codazzi equation;
Chapter 4 Minimal Surfaces: Isometric Deformations; Minimum conjugate surfaces; Examples
Chapter 5 Quaternions: Algebra; Rotation
Chapter 6 Differentiable Varieties: Examples; Covariant derivative; Riemannian metric; Geodesics and exponential application; Gauss-Bonnet theorem
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Main Bibliography |
1. Modern Differential Geometry of Curves and Surfaces with Mathematica 3rd Edition, Alfred Gray, Elsa Abbena and Simon Salamon, Studies in Advanced Mathematics, 2006
2. Elementary Differential Geometry, Andrew Pressley, Second Edition, Springer, 2012
3. Geometria Diferencial de Curvas e Superfícies, Manfredo do Carmo, Sociedade Brasileira de matemática, 6º edição, 2014
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Teaching Methodologies and Assessment Criteria |
In classes, the main results are presented and they are applied in the presentation of examples and solving exercises. The assessment consists of written tests with problems about these results, examples and exercises.
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Language |
Portuguese. Tutorial support is available in English.
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