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Lecturer(s)
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Peška Patrik, RNDr. Ph.D.
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Mikeš Josef, prof. RNDr. DrSc.
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Course content
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1. Vector functions. 2. Methods of definitions of curves. 3. Length of a curve, natural parameter. 4. Frenet's reper and equations. 5. Contact of curves, osculation circle. 6. Methods of definitions of surfaces. 7. Tangent plane and normal of surface. 8. First and second fundamental forms of the surface. Meussnier theorem. 9. Curvature of surface. Euler's formula. 10. Gauss and Weiengarten formulas, Egregium theorem. 11. Special curves on the surface. 12. Special surfaces. 13. Differentiaable manifolds, affine connection, Riemann manifolds.
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Learning activities and teaching methods
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Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration, Activating (Simulations, Games, Dramatization), Group work
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Learning outcomes
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The aim of the course is to introduce the fundamental concepts and methods of Differential Geometry of curves and surfaces in Euclidean space. Students will: - understand different ways of representing curves and surfaces, - master the Frenet frame and the basic invariants of curves, - acquire the ability to compute and interpret curvature of curves and surfaces, - understand the geometric meaning of tangent planes, normal vectors, and fundamental forms, - become familiar with key results such as Meusnier's theorem, Euler's formula, and Gauss's Theorema Egregium, - develop the ability to analyze special curves and surfaces, - gain a basic understanding of differentiable manifolds and affine connections.
After completing the course, the student: Knowledge: - understands the basic concepts of Differential Geometry of curves and surfaces, - knows the properties of important types of curves on surfaces (principal, asymptotic, geodesic), - understands key geometric quantities (curvature, normal vector, tangent plane, fundamental forms). Skills: - is able to work with different representations of curves and surfaces, - can compute curvature of curves and surfaces and interpret its geometric meaning, - applies the Frenet frame and related formulas in practical problems, - analyzes properties of special curves and surfaces. Competences: - is able to solve basic problems in differential geometry independently, - can connect geometric intuition with analytical methods, - understands the relationship between local and global properties of geometric objects.
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Prerequisites
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Knowledge of Analytic Geometry and Mathematical Analysis is assumed, in particular functions of several variables, differentiation, and integration. Basic knowledge of linear algebra is an advantage.
KAG/AGN
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Assessment methods and criteria
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Written exam, Student performance, Written exam
To successfully complete the course, the student must: - demonstrate understanding of the basic concepts of Differential Geometry of curves and surfaces, - perform computations involving the Frenet frame, curvature, and torsion of curves, - determine tangent planes, normal vectors, and key properties of surfaces, - work with the first and second fundamental forms, - analyze geodesic, principal, and asymptotic curves on surfaces, - solve both practical and theoretical problems, - complete continuous assessment tasks (assignments/tests), - pass the final examination.
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Recommended literature
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Budinský B., Kepr B. (1970). Základy diferenciální geometrie s technickými aplikacemi. Praha.
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. Brno.
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Gray A. (1994). Differential geometry.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. Olomouc.
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J. Mikeš, M. Sochor. (2013). Diferenciální geometrie ploch v úlohách. UP OLomouc.
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Metelka, J. (1969). Diferenciální geometrie. Praha.
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