G1011224 - Curvas e Superficies (Xeometría Diferencial) - Curso 2011/2012
- Créditos ECTS
- Créditos ECTS: 6.00
- Total: 6.0
- Horas ECTS
- Clase Expositiva: 42.00
- Clase Interactiva Laboratorio: 16.00
- Horas de Titorías: 2.00
- Total: 60.0
- Tipo: Materia Ordinaria Grao RD 1393/2007
- Departamentos: Xeometría e Topoloxía
- Áreas: Xeometría e Topoloxía
- Centro: Facultade de Matemáticas
- Convocatoria: 2º Semestre de Titulacións de Grao/Máster
- Docencia e Matrícula: null
Existen programas da materia para os seguintes idiomas:CastelánGalegoInglésCourse objectives
To use the differential and integral calculation and the Euclidean topology for the study of curves and surfaces in the Euclidean space 3-dimensional. To handle the method of the trihedral mobile (trihedral of Frenet) for the study of the local theory of curves. To be able to calculate lengths of curves, the curvature and the twist. To be able to work with regular surfaces through their coordinates. To recognize the nature of the points of a surface in the space. To know and to be able to calculate the normal and main curvatures of a surface, as well as the total curvature and the medium curvature. To use the acquired concepts to work with the ruled and minimal surfaces.
To use the software and computer means necessary to be able to visualize the curves and surfaces and calculate their elements.Contents
1.-Curves in the Euclidean space 3-dimensional Concept of curve. Analytical expressions. Singular points and regular points. Change parameter. Length of an arch of curve. Parameter length of arch.
2. - Frenet's formulae
Frenet´s trihedral. Frenet's formulae. Expressions of the curvature and the torsion.
3.-Center and radius of curvature
Centre and radius of curvature. Osculator circle.
Helix. Lancret's theorem. Characterization of helixes. Geometric considerations. Examples of helixes. Circular helixes.
5.-Curves defined from other Bertrand's Curves.
Spherical curves. Spherical Indicatrixes
7.-Basic theorem of curved
Linear transformations. Translations. Related transformations. Isometries and rigid movements. Orientation. Basic theorem.
Basic definitions. Examples. change parameters. Distinguishable functions on surfaces. The tangent plane. Differential of an application
9.-The first fundamental form of a surface. Applications
10.-The geometry of Gauss's application.
The second fundamental form of a surface. Normal curvatures. Theorems of Meusnier and Euler. Lines of curvature. Classification of points of a surface. Dupin's Indicatrix. Conjugate directions.
11.-Gauss's application in coordinate local Equations of Gauss and Weingarten. Differential equations of the asymptotic lines and curvature lines.
Codazzi-Mainardi's equations. Egregium Gauss Theorem . Bonnet's theorem.
13.-Practical applications Surfaces of revolution. Ruled surfaces. Developed surfaces. Basic and complementary bibliography
Araújo, P.V. Geometria Diferencial. Coleçao Matemática Universitária. IMPA, Rio de Janeiro. 1998.
Carmo, M.P.do. Geometría diferencial de curvas y superficies. Alianza ed. Madrid 1990.
Cordero, L.A., Fernandez, M., Gray, A. Curvas y superficies con Matemática. Addison-Wesley Iberoamericana. 1994.
Fedenko, A. Problemas de geometría diferencial. Mir. Moscú 1981.
Hsiung, C. C. A first course in differential geometry. Wiley. New York 1981.
Klingenberg, W. Un curso de geometría diferencial. Alhambra ed. Madrid 1973.
Lipschutz, L.M. Geometría diferencial. Schaum. Colombia 1971.
López de la Rica, A; de la Villa Cuenca, A. Geometría diferencial. Edit. Clagsa, Madrid 1997.
Milman, R.S., Parker, G.D. Elements of differential geometry. Prentice Hall.New J.1977.
Vaisman, I. A first course in differential geometry. Marcel Dekker.New York 1984.Competence
- To identify the regular curves, isolating singularities.
- Knowledge and managing of the curvature and the torsion of a regular curve through the Frenet's trihedral .
- Identification of abstract surfaces and regular surfaces.
- Using Gauss's application to the local study of a regular surface.
- Knowledge of the normal curvatures of a surface, of the principal curvatures and Gauss curvature and mean curvature.
- Using the previous thing for the study of surfaces known (surfaces of revolution, ruled and minimal).
- Using computer packages for the visualization of surfaces and calculation of their elements.Teaching methodology
The teaching of this subject will be developed on the following way:
42 hours of blackboard classes (on-site work at the classroom) where it is showed to the student the basic concepts and main theorems of the program. The remaining theorems, as well as diverse exercises will be raised to the student for their personal work relying on the support of 13 hours of tutorials in small groups. In addition, very punctual questions will be solved in two hours of tutorials in very small or individual groups. Finally, they spend three hours to the calculation of curvatures using computer programs.Assessment system
The grade of every student will be through continuous assessment and the accomplishment of a final examination. The continuous assessment will be done by means of written controls, works, participation of the student in the classroom and tutorials.
The grade of students will neither be lower than that of the final examination nor to the one obtained considering it with the continuous assessment, giving the latter a weight of 35 %.Study time and individual work
The personal work of students, without considering the on-site work at classroom, it is estimated in 60 hours. The writing of exercises, conclusions or other works 27 hours. Programming or other works in computers 3 hours. Total 90 hoursRecommendations for the study of the subject
Subjects that are advised to study previously:
Linear and multilinear Algebra, Topology of the Euclidean spaces, Differentiation of functions of several real variables To have studied or studying at this moment Introduction to the differential ordinary equations.Comments
The daily work is very important to follow the development of this constructive and intuitive subject.