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Lecturer(s)
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Mikeš Josef, prof. RNDr. DrSc.
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Course content
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1. Vector functions 2. Parametrization of curves. Orientation. Means of determinacy of curves. 3. Length of a curve, natural parameter. 4. Tangent, osculating plane, moving Frenet frame 5. Frenet formulae, curvature, torsion. Natural equations of a curve. Conditions for differentiability. 6. Contact of curves, osculating circle 7. Conics 8. Parametrization of surfaces. Means of determinacy of surfaces. 9. Tangent, tangential plane, normal of a surface 10. First and second fundamental form of a surface. 11. Messnier formulae and theorem 12. Principal directions. Normal, geodetic, principal, mean, Gauss curvature. Euler formulae. 13. Gauss and Weiengarten formulae. 14. Gauss and Peterson-Codazzi-Mainardi formulae. Christoffel symbols. 15. Theorem Egregium. 16. Special curves on a surface. 17. Special surfaces (developable, constant curvature, rotational) 18. Surfaces of the second order 19. Differentiable manifold, affine connection, Riemannian manifold 20. Variation problem, geodesics. Isoparametric curves.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Understand theory of curves and surfaces.
1. Knowledge Understanding of principles of the diffential geometry on curves and surfaces.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Credit: active participation on laboratory and succesfull to pass test. Exam: the student has to undrestand the subject and be able to prove the principal results.
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Recommended literature
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Berger, M. (1987). Geometry I, II. Universitext Springer-Verlag Berlin.
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Budínský, B., Kepr, B. (1970). Základy diferenciální geometrie s technickými aplikacemi. SNTL Praha.
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
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Gray, A. (1994). Differential geometry. CRC Press Icn.
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Gray, A. (2006). Modern Differential Geometry of Curves and Surfaces.. Chapman \& Hall/CRC, Boca Raton, FL.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. UP Olomouc.
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J. Mikeš, M. Sochor. (2013). Diferenciální geometrie ploch v úlohách. UP OLomouc.
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Kolář, I., Pospíšilová, L. (2007). Diferenciální geometrie křivek a ploch. El. publ. MU Brno.
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Metelka, J. (1969). Diferenciální geometrie. SPN Praha.
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Mikeš, J., Kiosak, V., Vanžurová, A. (2008). Geodesic mappings of manifolds with affine connection. UP Olomouc.
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Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
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