| Course title | Differential Geometry |
|---|---|
| Course code | KAG/ZG2 |
| Organizational form of instruction | Lecture + Exercise |
| Level of course | Master |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 3 |
| Language of instruction | Czech |
| Status of course | Compulsory-optional |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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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.
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| Learning activities and teaching methods |
| Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration, Activating (Simulations, Games, Dramatization) |
| 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. |
| 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/ZG ----- or ----- KAG/ZG1 ----- or ----- KAG/ZPG |
| Assessment methods and criteria |
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Oral exam, Written exam, Student performance
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. |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Science | Study plan (Version): Teaching Training in Mathematics for Secondary Schools (2019) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |