# PřF:M4190 Curves and Surfaces - Course Information

## M4190 Differential Geometry of Curves and Surfaces

**Faculty of Science**

Spring 2025

**Extent and Intensity**- 2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
**Teacher(s)**- doc. Mgr. Josef Šilhan, Ph.D. (lecturer)
**Guaranteed by**- doc. Mgr. Josef Šilhan, Ph.D.

Department of Mathematics and Statistics – Departments – Faculty of Science

Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science **Prerequisites**- (
**M2110**Linear Algebra II && (**M1100**Mathematical Analysis I ||**M1100F**Mathematical Analysis I ))||**M3501**Mathematical Analysis 3

The basic knowledge of the differential and integral calculus and analytic geometry is expected. **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Mathematics (programme PřF, B-MA)

**Course objectives**- A rounded survey of the main ideas and results of the differential geometry of curves and surfaces in the Euclidean three-space is presented. This covers both the theory of local invariants and also global properties of curves and surfaces.
**Learning outcomes**- At the end of this course, students should be able to apply differential geometry in mathematical analysis and applied mathematics.
**Syllabus**- Parametric expressions and equations of curves and surfaces. Contact of two curves and of a curve with a surface. Arc of a curve, Frenet frame, curvature and torsion of spacial curves. Envelops. The first and the second fundamental form of a surface, mean and Gauss curvatures. Covariant derivative, intrinsic and extrinsic geometry of surfaces. Gauss-Bonnet theorem.

**Literature**- KOLÁŘ, Ivan and Lenka POSPÍŠILOVÁ. Diferenciální geometrie křivek a ploch.
*Elportál*. Brno: Masarykova univerzita, 2008. ISSN 1802-128X.*URL*info

*recommended literature*- GRAY, Alfred.
*Modern differential geometry of curves and surfaces with mathematica*. 2nd ed. Boca Raton: CRC Press, 1997, xxiv, 1053. ISBN 0-8493-7164-3. info

*not specified*- KOLÁŘ, Ivan and Lenka POSPÍŠILOVÁ. Diferenciální geometrie křivek a ploch.
**Teaching methods**- Two hours of lectures and two hours of problem class.
**Assessment methods**- Exam written (focused on computation) and oral (focused on theoretical knowledge). At least 50% of the writen exam and at least basic theoretical knowledge on the oral exam is needed to pass.
**Language of instruction**- Czech
**Follow-Up Courses****Further Comments**- The course is taught annually.

The course is taught: every week.

- Enrolment Statistics (Spring 2025, recent)
- Permalink: https://is.muni.cz/course/sci/spring2025/M4190