| Code |
14776
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| Year |
3
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Real Analysis and Linear Algebra.
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Learning outcomes |
On completion of this unit, students should:
1. Parametrize regular plane curves and space curves and determine their curvature and torsion;
2. Classify curves through their curvature and torsion;
3. Recognize and parameterize regular surfaces in space and determine their curvature using local coordinates;
4. Determine the equations of the geodesics of a regular surface;
5. Classify special subclasses of surfaces.
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Syllabus |
1. Regular curves and parameterized curves. Plane and spatial curves parameterized by arc length. Curvature and torsion. Regular curves. Fundamental theorem of curves.
2. Regular surfaces. Parameterized surfaces and regular surfaces. First and second fundamental form. Orientable surfaces. Gauss application. Isometries and conformal maps. Gauss theorems and Bonnet's theorem.
3. Geodesics and curvature. Gaussian curvature and mean curvature. Geodesics and exponential application. Gauss Bonnet's theorem.
4. Minimal surfaces and ruled surfaces. Ruled and developable surfaces. Surfaces that minimize area. Examples of minimal surfaces.
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Main Bibliography |
1. Geometria diferencial de curvas e superfícies, Manfredo do Carmo, Sociedade Brasileira de matemática, 6º edição, 2014;
2. Elementary Differential Geometry, Andrew Pressley, Springer, 2001;
3. Modern Differential Geometry of Curves and Surfaces with Mathematica 3rd Edition, Alfred Gray, Elsa Abbena and Simon Salamon, Studies in Advanced Mathematics, 2006.
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Teaching Methodologies and Assessment Criteria |
- Theoretical-practical classes and homework problems;
- Great focus will be given to the construction and graphical visualization of example:
- The unit is assessed by two written tests (95%) and by problem-solving assignments (5%).
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Language |
Portuguese. Tutorial support is available in English.
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