| 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; Application of Gauss; Applications on surface and tangent vectors; Orientable surface; First Fundamental Form; Weingarten application; Normal curvature, average curvature and Gaussian curvature; Second Fundamental Form; Ruled and revolved surfaces; Minimum surface; Gauss's Egregium Theorem; Geodesic and normal curvatures of a curve; Geodesics; Umbilical points; Mainardi-Codazzi equation;
Chapter 4 Minimum 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 |
During Teaching-Learning, two tests will be carried out (F1 and F2) and two homework/exercises (T1 and T2), all evaluated at 20 points.
The final classification in Teaching-Learning will be obtained by the formula EA=0.4*(F1+F2)+.01*(T1+T2).
The student must attend at least 50% of classes to be admitted to the Exam.
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Language |
Portuguese. Tutorial support is available in English.
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