Course: Selected Chapters of Geometry

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Course title Selected Chapters of Geometry
Course code KAG/MDIG7
Organizational form of instruction Lecture + Exercise
Level of course Master
Year of study not specified
Semester Winter
Number of ECTS credits 3
Language of instruction Czech
Status of course Compulsory
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. Methods of determination of curves. 3. Length of a curve, natural parameters. 4. Tangents, planes of osculatory, the Frenet frame. 5. The Frenet formulas, curvature, torsion. Natural equation of a curve. 6. Joint of curves, circle of osculatory. 7. Parametrization of surfaces. Methods of determination of surfaces. 8. Tangents. Tangent planes and normals of a surface. Orientation of surfaces. 9. First and second fundamental form of a surface anf their purpose. 10. The Meussnier formulas and theorem. 11. Principal directions. Normal, geodetic, principal, medium and Gauss curvatures. Euler's formula. 12. Gauss and Weiengarten formulas. 13. Gauss and Peterson-Codazzi-Mainardi formulas. Christoffel symbols. 14. The Egregium theorem. 15. Special curves on surfaces. 16. Special surfaces (set surfaces, surface of a constant curvature, surfaces of revolution). 17. Differentiable manifolds, affine connections, the Riemann manifolds.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
Learning outcomes
Describe principles on differential geometry on curves, surfaces and manifold.
1. Knowledge Describe properties of the differential geometry of curves and surfaces and manifolds.
Knowledge of principles of analytical geometry.

Assessment methods and criteria
Oral exam, Written exam

Credit: the student has to participate actively in seminars and pass the test. Exam: oral. The student has to understand the subject and be able to prove the principal results.
Recommended literature
  • Budinský 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 Inc.
  • Metelka, J. (1969). Diferenciální geometrie. SPN Praha.
  • 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 Teaching Training in Mathematics for Secondary Schools (2015) Pedagogy, teacher training and social care 1 Winter