Course: Differential Geometry

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Course title Differential Geometry
Course code KMA/DG
Organizational form of instruction Lecture + Seminar
Level of course Bachelor
Year of study 2
Semester Winter and summer
Number of ECTS credits 5
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Ježek František, Doc. RNDr. CSc.
  • Tomiczková Světlana, RNDr. Ph.D.
Course content
1. Curve equations, parameterization. 2. Frenetovy curvature formula for a curve. 3. Osculating circle, canonical and natural form, envelope curve. 4. Surface equations, parameterization. 5. Tangent properties, envelope surface. 6. Tangent properties, developable surface. 7. Vectors and tensors on surface: covariant and contravariant coordinates, order of tensor, first fundamental tensor. 8. The first fundamental form of surfaces: the length of the curve on the surface, projection and developing surfaces, conformal mapping. 9. The second fundamental form of surfaces: normal surface curvature, Meusnier proposition, Dupin indikatrix. 10. Mean and Gaussian curvature: the main and the asymptotic curves, Weigarten equation. 11. Geodetic curvature, geodesic. 12. Christoffelovy symbols, the absolute derivative of a vector field. 13. Minimal space: definition, properties and examples. 14. Differential manifolds, summary.

Learning activities and teaching methods
Lecture, Practicum
  • Contact hours - 52 hours per semester
  • Preparation for comprehensive test (10-40) - 30 hours per semester
  • Presentation preparation (report) (1-10) - 10 hours per semester
  • Preparation for an examination (30-60) - 50 hours per semester
prerequisite
professional knowledge
We require a basic orientation in the concepts and skills of differential and integral calculus of functions of one or more real variables. Basic orientation in the foundations of linear algebra and analytic geometry is also necessary.
professional skills
aplikovat dovednosti získané úspěšným zvládnutím základních kurzů matematiky (matematické analýzy) a lineární algebry
teoreticky i prakticky pracovat s pojmy derivace a integrál
řešit soustavy lineárních algebraických rovnic
pracovat s lineárními a kvadratickými objekty (rovnice, společné body)
pracovat s vektory (lineární závislost, vektorový, skalární, smíšený součin)
general eligibility
N/A
N/A
N/A
N/A
learning outcomes
professional knowledge
The student is able to describe curves and surfaces in parametric form (and vice versa, the parametric description to visualize) and from this description to derive important characteristics of the object, especially its curvature.
professional skills
pro křivku a plochu vytvořit parametrický popis
z parametrického popisu provést vizualizaci objektu (křivky a plochy)
z popisu odvodit důležité charakteristiky křivky, zejména jeho křivosti (křivost a torze)
z popisu odvodit důležité charakteristiky plochy (1. a 2. tenzor plochy, Gaussova, střední a geodetická křivost)
ze zjištěných charakteristik odvodit vlastnosti popř. typ křivek (rovinnost, obalové křivky, tečné vlastnosti)
ze zjištěných charakteristik odvodit vlastnosti popř. typ ploch (rozvinutelné plochy, minimální plochy, obalové plochy, křivky na plochách)
general eligibility
N/A
teaching methods
professional knowledge
Lecture supplemented with a discussion
Practicum
professional skills
Lecture supplemented with a discussion
Practicum
general eligibility
Lecture supplemented with a discussion
Practicum
assessment methods
professional knowledge
Combined exam
professional skills
Combined exam
general eligibility
Combined exam
Recommended literature
  • Budinský, B. Analytická a diferenciální geometrie. 1. vyd. Praha : SNTL, 1983.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Education Biology in Education (16) Biology courses 2 Summer
Faculty of Education Physics in Education (16) Physics courses 2 Summer
Faculty of Education Studies in Mathematics (13) Mathematics courses 2 Summer
Faculty of Education Chemistry in Education (17) Chemistry courses 2 Summer
Faculty of Education - (17) Pedagogy, teacher training and social care 2 Summer
Faculty of Applied Sciences Geomatics (2012) Construction industry, geodesy and cartography 3 Summer
Faculty of Education Studies in Mathematics (15) Mathematics courses 2 Summer
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 2 Summer
Faculty of Applied Sciences Mathematics for Science (2012) Mathematics courses 2 Summer
Faculty of Education Chemistry in Education (16) Chemistry courses 2 Summer
Faculty of Education - (16) Pedagogy, teacher training and social care 2 Summer
Faculty of Education Studies in Mathematics (17) Mathematics courses 2 Summer
Faculty of Education Physics in Education (17) Physics courses 2 Summer
Faculty of Applied Sciences Geomatics (2014) Construction industry, geodesy and cartography 3 Summer
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 2 Summer
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 2 Summer
Faculty of Education Information Technologies in Education (16) Informatics courses 2 Summer
Faculty of Education Geography in Education (16) Pedagogy, teacher training and social care 2 Summer
Faculty of Applied Sciences Geomatics (2016) Construction industry, geodesy and cartography 3 Summer
Faculty of Applied Sciences Geomatics (2016) Construction industry, geodesy and cartography 3 Summer
Faculty of Education Studies in Mathematics (14) Mathematics courses 2 Summer
Faculty of Education Information Technologies in Education (17) Informatics courses 2 Summer
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 2 Summer
Faculty of Applied Sciences Scientific Computing and Modelling (2011) Mathematics courses 2 Summer
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 2 Summer
Faculty of Education Studies in Mathematics (1) Mathematics courses 2 Summer
Faculty of Education Studies in Mathematics (16) Mathematics courses 2 Summer
Faculty of Education Geography in Education (17) Pedagogy, teacher training and social care 2 Summer
Faculty of Applied Sciences Geomatics (2017) Construction industry, geodesy and cartography 3 Summer
Faculty of Applied Sciences Geomatics (2014) Construction industry, geodesy and cartography 3 Summer
Faculty of Applied Sciences Geomatics (2017) Construction industry, geodesy and cartography 3 Summer
Faculty of Education Biology in Education (17) Biology courses 2 Summer
Faculty of Education Physical Education in Education (17) Physical education and sport 2 Summer
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 2 Summer
Faculty of Applied Sciences Geomatics (2015) Construction industry, geodesy and cartography 3 Summer
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 2 Summer
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 2 Summer