SylabUZ
Nazwa przedmiotu | Geometria |
Kod przedmiotu | 11.1-WK-IiEP-G-Ć-S14_pNadGen83WZF |
Wydział | Wydział Matematyki, Informatyki i Ekonometrii |
Kierunek | Computer science and econometrics |
Profil | ogólnoakademicki |
Rodzaj studiów | pierwszego stopnia z tyt. licencjata |
Semestr rozpoczęcia | semestr zimowy 2019/2020 |
Semestr | 5 |
Liczba punktów ECTS do zdobycia | 5 |
Typ przedmiotu | obieralny |
Język nauczania | polski |
Sylabus opracował |
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Forma zajęć | Liczba godzin w semestrze (stacjonarne) | Liczba godzin w tygodniu (stacjonarne) | Liczba godzin w semestrze (niestacjonarne) | Liczba godzin w tygodniu (niestacjonarne) | Forma zaliczenia |
Wykład | 30 | 2 | - | - | Egzamin |
Ćwiczenia | 30 | 2 | - | - | Zaliczenie na ocenę |
The course has to main goals: developing skills of ‘geometrizing’ mathematical problems, solving geometric problems by algebraic methods.
Linear algebra 2.
Lecture
Affine and Euclidean point spaces.
1. Affine combination of points; affine independency; examples of affine spaces; isomorphisms of affine spaces; a standard model of an affine space. Affine mappings. (4h)
2. Affine subspaces: hyperplane, line. Particular subsets of an affine space: line segments, convex sets, simplices. Sets of points in a general postion. Convex hull, polytope as a convex hull of a finite set. Caratheodory’s theorem. Radon’s theorem, Helly’s theorem. (6h)
3. Euclidean point spaces: distance, ball, classification of isometries. (2h)
4. Halfspaces: geometric interpretation of linear inequations. Paralleotopes, cubes. (2h)
5. Closed convex sets; the distance of a point from a convex set and a hyperplane. (2 godz.)
6. Volume of a set – volume of a parallelotope and a simplex; Brunn–Minkowski inequality; John ellipsoid. (8h)
Projective spaces
1. Definition, basic properties, projective mappings(2h)
Quadric surfaces
1. Classification of conics and general quadrics.(4h)
Class
1. Exercises in elementary geometry (4h)
2. Elements of spherical geometry, spherical polytopes (formulas to be derived as exercises). Euler’s formula for convex and spherical polytopes. Applications. (4h)
3. Compositions of isometries of the plane and the space. (3h)
4. Applications of Helly’s theorem. (2h)
5. Finding the distance from a point to a set. (2h)
6. Finding the Minkowski’s sum of convex figures and estimation of the volume of the sum – isoperimetric inequality. (2)
7. Minkowski’s theorem on lattice points (the proof as a series of exercises); applications (2h)
8. Discussion over essays. (2h)
9. Informal introduction to the Euler characteristics– counting the Euler characteristic of selected set (e.g. closed surfaces). (2h)
10. Properties of conics and quadric surfaces (4h)
11. Class test (2h)
Traditional lecturing, solving problems under the supervision of the instructor, preparing presentations or essays (collaborative effort).
Opis efektu | Symbole efektów | Metody weryfikacji | Forma zajęć |
1. Preparation of the students and their active participation is assessed during each class by their instructor.
2. Class test with problems of diverse difficulty helping to assess whether a student achieved minimal outcomes.
3. Essay or presentation prepared by a team.
4. Written examination: It consists of three theoretical problems (sth need to be proved or explained) and four practical problems (sth need to be calculated, checked or found).
Final grade = 0.4 x class grade + 0,6
1. M. Berger, Geometry I and II, Universitext, Springer.
1. H. Hopf, Differential Geometry in the Large, LNM 1000, Springer, 1989.
2. J. Matoušek, Lectures on discrete geometry, Springer, 2002.
3. M. Aigner, G. M. Ziegler, Proofs from the BOOK, Springer 2004.
Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 21-11-2020 06:10)