Course detail

Differential Geometry

FSI-SDGAcad. year: 2025/2026

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.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, analytic geometry, differential and integral calculus of functions of one and several variables.

Rules for evaluation and completion of the course

Active attendance at the seminars and written test.
In a 120-minute written test, students have to solve assigned problems.
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.

Aims

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. Pressley: Elementary Differential Geometry, Springer- Verlag, 2012 (EN)
K. Tapp: Differential Geometry of Curves and Surfaces, Springer-Verlag, 2016 (EN)
M. A. Akivis, V. V. Goldberg: An Introduction to Linear Algebra and Tensors, Dover Publications, New York, 1972 (EN)
M. Umehara, K. Yamada: Differential Geometry of Curves and Surfaces, World Scientific, 2015 (EN)
Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (Prentice Hall, Inc. 1976) (EN)

Recommended reading

Boček L.: Tenzorový počet (SNTL Praha) (CS)
I. Kolář, L. Pospíšilová: Diferenciální geometrie křivek a ploch, elektronické skriptum MU (CS)
M. Doupovec : Diferenciální geometrie a tenzorový počet (skriptum VUT) (CS)
M. P. Do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 (EN)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 2 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.

Syllabus

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

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miroslav Doupovec, CSc., dr. h. c.

Syllabus

Seminars related to the lectures given in the previous week.

E-learning texts

Miroslav Doupovec: Diferenciální geometrie a tenzorový počet, skriptum VUT, 1999.
K0.pdf 0.43 MB
K1.pdf 0.26 MB
K2.pdf 1.26 MB
K3.pdf 2.88 MB
K4.pdf 0.18 MB