The aim of studying and bibliography

The main aim of study: Getting knowledge about the notions and quantities characterizing curves and surfaces. Getting ability of classifying curves and surfaces and determining their geometric invariants such as curvatures.

The general information about the module: The module concerns a theory of spatial curves and surfaces. The theory of curves cntains the arclength parameter, the Frenet frame, fundamental theorems and theorems involving invariants. The surface theory leads to determining various curvatures (Gauss, mean, normal curvature) and classifying points of a surface. The Riemanian metric and properties of curves contained in a surface are also studied.

Bibliography required to complete the module

Bibliography used during lectures

1	John Oprea	Geometria różniczkowa i jej zastosowania	PWN.	2002
2	Jacek Gancarzewicz, Barbara Opozda	Wstęp do geometrii różniczkowej	Wydawnictwo Uniwersytetu Jagiellońskiego.
3	Biogusław Gdowski	Elementy geometrii różniczkowej z zadaniami	Oficyna Wydawnicza Politechniki Warszawskiej.	2005
4	Theodore Shifrin	Differential Geometry: A First Course in Curves and Surfaces	http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf.	2016

Bibliography used during classes/laboratories/others

1	Bogusław Gdowski	Elementy geometrii różniczkowej z zadaniami	Oficyna Wydawnicza Politechniki Warszawskiej.	2005
2	A.N.Pressley	Elementary Differential Geometry	Springer.	2010
3	Theodore Shifrin	Differential Geometry: A First Course in Curves and Surfaces	http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf.	2016

Bibliography to self-study

1

Manfredo Do Carmo

Differential Geometry of Curves and Surfaces

Pearson.

1976

Basic requirements in category knowledge/skills/social competences

Formal requirements: Requirements accordant with Rules and Regulations of studies

Basic requirements in category knowledge: Knowledge of analysis of several variables, ODE, especially systems of ODE, linear algebra.

Basic requirements in category skills: Ability of calculating integrals, differentiating mappings of several variables, calculating eigenvalues and eigenvectors of matrices.

Basic requirements in category social competences: Ability of individual and group learning.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with OEK
01	A student determines Frenet reper of a regular space curve and its curvature and torsion.	lecture, classes	writing test	K_W01++ K_U16+	X2A_U02 X2A_U04 X2A_U06
02	Lists and explains theorems characterizing curves lying in a plane or within a circle. Uses the theorems to analyze given curves. States the fundamental theorems	lecture, problems classes	writing test, oral exam	K_W01++ K_W02+++ K_W03+ K_W04++ K_W05++ K_K02++	X2A_W02 X2A_W03 X2A_K01 X2A_K02
03	Student can assess whether a surface patch is regular, can determine principal curvatures using the shape operator, determines the Gauss curvature of the surface.	lecture, classes	writing test	K_W01+ K_W03+ K_U10+++	X2A_U01
04	Student classifies points of a surface.	lecture, problems classes	writing test. oral exam	K_W02++ K_K02++	X2A_W03 X2A_K01 X2A_K02
05	Tells the definitions and theorems included in the lecture.	lecture, problems classes	oral or written exam	K_W01+++ K_W03+

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Regular space curves, various parametrizations, the arclength parameter, the curvature and torsion, Frenet equations and frame, Characterization of planar curves and curves contained in a circle. Spherical curves.	W01, W02,W03, W04, C01 - C04	MEK01 MEK02
3	TK02	Frenet frame for an arbitrary parametrization. Existence and rigidity theorems.	W05,W06, C05,C06,C07	MEK01
3	TK03	A regular surface patch. Applyig of Implicit Funkction Theorem. The tangent space and the normal field. Orientation. Curves on a surface. The first fundamental form and the Riemannian metric.	w07-w09, C08-C10	MEK03 MEK04
3	TK04	The shape operator, the normal curvature. The Lagrange identity. The Gauss curvature and the mean curvature. The main direciions and curvatures. The second fundamental form. Geodesics. Minimal surfaces.	W10-W13, C11-C13	MEK03 MEK04
3	TK05	Manifoldsin of arbitrary dimension, atlas, tangent space, Riemannian metric.	w14-w15, c14-c15	MEK05

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 10.00 hours/sem.	contact hours: 30.00 hours/sem.	complementing/reading through notes: 15.00 hours/sem. Studying the recommended bibliography: 15.00 hours/sem.
Class (sem. 3)	The preparation for a Class: 20.00 hours/sem. The preparation for a test: 5.00 hours/sem.	contact hours: 30.00 hours/sem.	Finishing/Studying tasks: 5.00 hours/sem.
Advice (sem. 3)
Exam (sem. 3)	The preparation for an Exam: 15.00 hours/sem.	The written exam: 2.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	Oral or written exam
Class	Presenting solving of problems, test, presentation.
The final grade	Degree accounts the exam, test and activity. Degree higher than 4,0 is possible only after having passed an oral exam consisting of giving proofs of selected theorems.

Sample problems

Required during the exam/when receiving the credit
mek01-04.pdf
zagadnienia_2014-15.pdf

Realized during classes/laboratories/projects
geometria1.odt
geometria2.odt

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: no