Continuation of Differential geometry I, focuses on deeper properties of curves and surfaces.
Last update: T_KDM (04.05.2012)
Přednáška navazuje na předmět Diferenciální geometrie I a prohlubuje znalosti křivek a ploch.
Aim of the course -
Last update: T_KDM (19.05.2008)
This course helps to obtain theoretical background for teaching mathematics at high school.
Last update: T_KDM (19.05.2008)
Předmět pomáhá získat teoretické zázemí pro vyučování matematiky na střední škole.
Literature -
Last update: doc. RNDr. Antonín Slavík, Ph.D. (02.10.2012)
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.
P. M. H. Wilson, Curved Spaces (From Classical Geometries to Elementary Differential Geometry). Cambridge University Press, 2008.
E. Kreyszig, Differential Geometry, New York, 1991.
Ch. Bär, Elementary Differential Geometry, Cambridge University Press, 2010.
B. O'Neill, Elementary Differential Geometry (2nd edition), Elsevier, 2006.
A. Pressley, Elementary Differential Geometry (2nd edition), Springer, 2010.
M. Spivak, A comprehensive introduction to differential geometry vol. 1-5. Publish or Perish, Inc.
J. Oprea, Differential Geometry and Its Applications, The Mathematical Association of America, 2007.
C. G. Gibson, Elementary Geometry of Differentiable Curves, Cambridge University Press, 2001.
Last update: doc. RNDr. Antonín Slavík, Ph.D. (02.10.2012)
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.
P. M. H. Wilson, Curved Spaces (From Classical Geometries to Elementary Differential Geometry). Cambridge University Press, 2008.
E. Kreyszig, Differential Geometry, New York, 1991.
Ch. Bär, Elementary Differential Geometry, Cambridge University Press, 2010.
B. O'Neill, Elementary Differential Geometry (2nd edition), Elsevier, 2006.
A. Pressley, Elementary Differential Geometry (2nd edition), Springer, 2010.
M. Spivak, A comprehensive introduction to differential geometry vol. 1-5. Publish or Perish, Inc.
J. Oprea, Differential Geometry and Its Applications, The Mathematical Association of America, 2007.
C. G. Gibson, Elementary Geometry of Differentiable Curves, Cambridge University Press, 2001.
Teaching methods -
Last update: T_KDM (20.05.2008)
Lectures and exercises.
Last update: T_KDM (20.05.2008)
Přednáška a cvičení.
Syllabus -
Last update: doc. RNDr. Antonín Slavík, Ph.D. (02.10.2012)
Plane curves: envelopes, isoperimetric inequality, Crofton's formula and its applications, four vertex theorem.
The equations of Gauss, Christoffel symbols, Theorema egregium. Geodesic curves on surfaces and their properties, geodesic polar coordinates and their applications, Minding's theorem. Geometry on curved surfaces (geodesic circles and triangles). Surfaces with constant Gaussian curvature, minimal surfaces.
Last update: doc. RNDr. Antonín Slavík, Ph.D. (02.10.2012)
Rovinné křivky: obálka soustavy křivek, konvexní křivky, izoperimetrická nerovnost, Croftonův vzorec a jeho aplikace, věta o čtyřech vrcholech.
Plochy: Gaussovy rovnice, Christoffelovy symboly, Theorema egregium. Geodetické křivky na ploše a jejich vlastnosti, geodetické polární souřadnice a jejich použití, Mindingova věta. Geometrie na zakřivených plochách (geodetické kružnice a trojúhelníky). Plochy s konstantní Gaussovou křivostí, minimální plochy.
