Differential geometry
Some basic information about the module
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
| 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 |
| 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 |
| 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 |
|---|---|---|---|---|---|
| MEK01 | A student determines Frenet reper of a regular space curve and its curvature and torsion. | lecture, classes | writing test |
K-W01++ K-U16+ |
U02 U04 U06 |
| MEK02 | 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++ |
W02 W03 K01 K02 |
| MEK03 | 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+++ |
U01 |
| MEK04 | Student classifies points of a surface. | lecture, problems classes | writing test. oral exam |
K-W02++ K-K02++ |
W03 K01 K02 |
| MEK05 | Tells the definitions and theorems included in the lecture. | lecture, problems classes | oral or written exam |
K-W01+++ K-W03+ |
The syllabus of the module
| Sem. | TK | The content | realized in | MEK |
|---|---|---|---|---|
| 3 | TK01 | W01, W02,W03, W04, C01 - C04 | MEK01 MEK02 | |
| 3 | TK02 | W05,W06, C05,C06,C07 | MEK01 | |
| 3 | TK03 | w07-w09, C08-C10 | MEK03 MEK04 | |
| 3 | TK04 | W10-W13, C11-C13 | MEK03 MEK04 | |
| 3 | TK05 | 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 |
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