Differential Geometry (FSI-SDG)

Academic year 2022/2023
Supervisor: prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The course aims to acquaint the students with the basics of classical differential geometry of curves and surfaces. Another goal of the course is to develop the students' logical thinking.
Learning outcomes and competences:
Students will be made familiar with classical differential geometry of curves and surfaces. They will be able to apply this theory in various engineering tasks.
Prerequisites:
Linear algebra, analytic geometry, differential and integral calculus of functions of one and several variables.
Course contents:
The classical differential geometry of curves and surfaces: contact of curves, Frenet formulas, osculating curves, contact of surfaces, the first and the second fundamental form, asymptotic curves, Gauss curvature, ruled surfaces, the intrinsic geometry of a surface. Elements of Tensor Calculus.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Active attendance at the seminars and written test.
In a 120-minute written test, students have to solve assigned problems.
Controlled participation in lessons:
Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is fully at the discretion of the teacher.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture Week 1: The notion of a curve.
Week 2: The contact of curves.
Week 3: Frenet formulas of a plane curve.
Week 4: Osculating curves.
Week 5: Frenet formulas of a space curve.
Week 6. The notion of a surface.
Week 7: The contact of surfaces.
Week 8: The first fundamental form.
Week 9: The second fundamental form.
Week 10: Asymptotic curves.
Week 11: The Gauss curvature.
Week 12: Ruled surfaces.
Week 13: The intrinsic geometry of a surface.
    Exercise Seminars related to the lectures given in the previous week.
Literature - fundamental:
1. M. A. Akivis, V. V. Goldberg: An Introduction to Linear Algebra and Tensors, Dover Publications, New York, 1972
2. Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (Prentice Hall, Inc. 1976)
3. A. Pressley: Elementary Differential Geometry, Springer- Verlag, 2012
4. M. Umehara, K. Yamada: Differential Geometry of Curves and Surfaces, World Scientific, 2015
5. K. Tapp: Differential Geometry of Curves and Surfaces, Springer-Verlag, 2016
Literature - recommended:
1. M. Doupovec : Diferenciální geometrie a tenzorový počet (skriptum VUT)
2. I. Kolář, L. Pospíšilová: Diferenciální geometrie křivek a ploch, elektronické skriptum MU
3. Boček L.: Tenzorový počet (SNTL Praha)
4. M. P. Do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
CŽV full-time study CZV Bases of Mechanical Engineering -- GCr 4 Compulsory 1 1 S
B-MAI-P full-time study --- no specialisation -- GCr 4 Compulsory 1 2 S