Differential Geometry

Course code

MLM6407.DT

old course code

Course title in Estonian

Diferentsiaalgeomeetria

Course title in English

Differential Geometry

ECTS credits

6.0

approximate amount of contact lessons

78

Teaching semester

autumn

Assessment form

Examination

lecturer of 2019/2020 Spring semester

õppejõud on määramata

lecturer of 2020/2021 Autumn semester

lecturer not assigned

Course aims

To teach students to model curves and surfaces of the three-dimensional space mathematically. To introduce the main differential geometric characteristics of manifolds and to teach the application process of those characteristics. To introduce main notions of differential geometry with their applications. To use modern computer technology for solving the exercises in differential geometry.

Brief description of the course

Vector-valued function of scalar variable. Equations of a smooth curve. Tangent of a curve, normal plane and osculating plane. Moving frame of a curve, Bartels-Frenet-Serret formulae. Curvature and torsion of a curve. Osculating circle, evolutes and evolvents. The notion and equations of a surface, tangent plane and normal. First fundamental form of a surface, metric magnitudes on the surface. Second fundamental form of a surface. Normal curvature of a surface. General directions of surfaces and lines of curvature. Asymptotic directions of a surface and asymptotic curves. Geodetic curvature of a curve, geodetic lines. Theorem of Gauss. Theorem of Gauss-Bonnet. Intrinsic geometry of surfaces of constant curvature. The fundamental theorem of surfaces.

Independent work

Independent work: independent work includes work with lecture notes and textbooks, solving exercises. Solving home works and formulating their solutions. For it is needed independent work (ca 20 hours) in the computer class. Attending lectures is a prerequisite of the learning process.

Learning outcomes in the course

The student knows the mathematical facts and methods introduced in the lectures, is able to justify and apply them

Assessment methods

Examination. Passing two classroom tests and presenting two homeworks by computer is a prerequisite for the right to participate at the exam. In the exam test there are two theoretical questions and one exercise.

Teacher

lekt Mati Väljas

Prerequisite course 1

Study literature

Lumiste, Ü. (1963) Diferentsiaalgeomeetria Tallinn : Valgus;

Lumiste, Ü. (1987). Diferentsiaalgeomeetria. Tallinn : Valgus;

Gray, A. (1998). Modern differential geometry of Curves and Surfaces with Mathematica. USA : CRC Press LLC.

Lumiste, Ü. (1987). Diferentsiaalgeomeetria. Tallinn : Valgus;

Gray, A. (1998). Modern differential geometry of Curves and Surfaces with Mathematica. USA : CRC Press LLC.