Course: Geometry 1

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Course title Geometry 1
Course code KAG/GEO1M
Organizational form of instruction Lecture + Exercise
Level of course Bachelor
Year of study not specified
Semester Summer
Number of ECTS credits 4
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Mikeš Josef, prof. RNDr. DrSc.
Course content
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.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
Learning outcomes
Understand theory of curves and surfaces.
1. Knowledge Understanding of principles of the diffential geometry on curves and surfaces.

Assessment methods and criteria
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.
Recommended literature
  • Berger, M. (1987). Geometry I, II. Universitext Springer-Verlag Berlin.
  • Budínský, B., Kepr, B. (1970). Základy diferenciální geometrie s technickými aplikacemi. SNTL Praha.
  • Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
  • Gray, A. (1994). Differential geometry. CRC Press Icn.
  • Kolář, I., Pospíšilová, L. (2007). Diferenciální geometrie křivek a ploch. El. publ. MU Brno.
  • Metelka, J. (1969). Diferenciální geometrie. SPN Praha.
  • Mikeš, J., Kiosak, V., Vanžurová, A. (2008). Geodesic mappings of manifolds with affine connection. UP Olomouc.
  • Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
  • Pogorelov, A. V. (1969). Diferencialnaja geometrija.. Nauka Moskva.
  • Vanžurová, A. (1996). Diferenciální geometrie křivek a ploch. UP Olomouc.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Mathematics-Economics of Banking / Insurance Systems (1) Mathematics courses 2 Summer
Faculty of Science Mathematics and Applications (1) Mathematics courses 1 Summer