Curves and Surfaces

To use the differential and integral calculation and the Euclidean topology for the study of curves and surfaces in the Euclidean plane and the 3-dimensional space. 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 Gauss 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.

1. - Curves in the three-dimensional Euclidean space (5 expository hours)
Parametrized curves. Reparametrization. Arc length. Rigid motions: congruent curves.

2. - Curvature and torsion. The Fundamental Theorem of the Theory of Curve (5 expository hours)
Curvature and torsion. Frenet formulas. The Fundamental Theorem.

3. - Regular surfaces (10 expositive expository hours)
Basic Definitions. Examples. Local parametrizations. Differentiable Funcions on surfaces. The tangent plane. The differential of a differentiable map.

4. - The first fundamental form (6 expositive expository hours)
The first fundamental form. Isometries and intrinsic geometry. Applications.

5. - The Gauss map (10 expositive expository hours)
The second fundamental form. Normal curvatures. Normal sections. The Gauss curvature and the mean curvature. Umbilical points

6. - The Egregium Theorem of Gauss (6 expository expository hours)
Compatibility equations of Codazzi-Mainardi. The Egregium Theorem of Gauss. The Fundamental Theorem of surfaces in the Euclidean space (Theorem of Bonnet).

Basic bibliography
M. do Carmo; Geometría Diferencial de curvas y superficies. Alianza Ed. Madrid 1990
A. Fedenko; Problemas de Geometría Diferencial. MIR. Moscú 1981
M. A. Hernández Cifre, J. A. Pastor González; Un curso de geometría diferencial. CSIC. Madrid 2010

Complementary bibliography
M. Abate, F. Tovena; Curves and Surfaces. Springer-Verlag 2012
A.F. Costa, J.M. Gamboa, A.M. Porto; Notas de Geometría Diferencial de Curvas y Superficies (Volúmenes I y II, Teoría y Ejercicios). Ed. Sanz y Torres 2005
L. A. Cordero, M. Fernández, A. Gray; Curvas y superficies con Mathematica. Addison- Wesley Iberoamericana 1994
L. M. Lipschutz; Geometría diferencial. Schaum. Colombia 1971
A. López de la Rica, A. de la Villa Cuenca; Geometría diferencial. Ed. Clagsa. Madrid 1997
S. Montiel, A. Ros; Curves and surfaces. Graduate Studies in Mathematics 69. Amer. Math. Soc., Providence, RI 2009.

GENERAL COMPETENCES

CX1.- To learn the most important concepts, methods and results of the different branches of mathematics, together with some historical perspective of their development.
CX3.- To apply practical and theoretical mathematical knowledge, as well as analytical and abstraction capabilities, in posing and solving problems in any professional or academic context.
CX4.- To communicate in oral and written form mathematical knowledge, ideas, methods and results to either specialists or non-specialists in the field.
CX5.- To study and learn autonomously, with good organization in the use of time and resources, new results and techniques in any scientific or technological discipline.

SPECIFIC COMPETENCES

CE1.- To understand and use the mathematical language.
CE2.- To understand the rigorous proofs of some classical mathematical theorems.
CE3.- To be able to provide proofs of mathematical results. To formulate conjectures, and to design strategies to prove them or to show that they are false.
CE4.- To identify mistakes or wrong arguments, and provide proofs or counterexamples.
CE5.- To efficiently grasp the content or definition of a new mathematical object, and be able to relate it/them with known objects and to use it/them in different contexts.
CE6.- To be able to identify and make abstractions of the fundamental issues of a given problem, and be able to distinguish them from those that are purely circumstantial.


TRANSVERSAL COMPETENCES

CT1.- To use appropriate references as well as searching tools and bibliographic mathematical resources, including internet browsing.
CT2.- To use in optimal form the working time and to organize the available resources by establishing priorities and alternative approaches, as well as identifying logic errors in any decision making process.
CT3.- To corroborate or to refute in a rational way others’ arguments.
CT4.- To be able to read science, both in mother tongue as well as in foreign language when the latter is relevant for the scientific issue under study; specially in English.

The general methodological indications set forth in the Memory of the Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.

Teaching is programmed in Expositive, Interactive and Tutorial classes (in very small groups).

Expositive Teaching: The lectures will be devoted to the presentation and development of the essential contents of the subject.

Interactive Teaching: Interactive classes will be devoted to presenting examples and solving problems (both theoretical and the scope of applications). Individual or group work will be organized and problems will be proposed to be solved by students. In the interactive teaching classes, maximun participation and implication of the students is required, as the discussion, debate and resolution of the proposed tasks, aim to practice and strengthen their knowledge and to work some of the aforementioned competences.

Tutorials: The tutorial sessions are designed especially to stimulate the activity of the students outside the classroom. These will be used so that interested students can continuously examine their learning process, and for teachers to do a direct follow-up of this learning, which will allow them to detect insufficiencies and difficulties that can be corrected as they occur.

The weekly distribution of the subject will be approximately the following: 3 hours of expository class, 1 hour of interactive class. Throughout the course there will be 2 hours of tutorials (in very small groups)

Without prejudice to the general evaluation criteria for all subjects of the Degree, for the calculation of the final grade, the qualification of the continuous assessment and the qualification of the final exam will be considered.

Continuous assessment (25%). One written control will be carried out.


Final exam (75%). A final written exam will be carried out, which will allow to verify the knowledge acquired in relation to the concepts and results of the subject and the capacity of its application to specific cases.
The written exam will consist of a part of theoretical questions which may include definition of concepts, statements of results or total or partial proof of them and another part that will be the resolution of exercises, which will be similar to the ones proposed during the semester.

The final grade will be the maximum of those corresponding to the final exam and the final exam grade weighted with the continuous evaluation.

In addition to the specific competences of the subject, the competences CX1, CX3, CX4, CE1, CE2, CE3, CE4, CE5, CE6 and CE6 will be evaluated.

The evaluation system will be the same for the two opportunities of the call.

In case of dishonest execution of exercises or assessments the following regulation will be applied:
https://www.xunta.gal/dog/Publicados/2011/20110721/AnuncioG2018-190711-….


THE EXAMS IN THE GROUPS WILL BE DIFFERENT BUT EQUIVALENT

Time in presence of the teacher:
Expository classes: 42 h.
Interactive classes of laboratory with computer: 2 h.
Interactive laboratory classes / tutorials in a small group: 12 h.
Tutorials in very small groups or individualized: 2 h.

Total hours of classroom work in class 58

Time of personal work:
Self study, individual or group: 55 h.
Writing of exercises, conclusions and other works: 25 h.
Programming / experimentation or other work in computer / laboratory: 2 h.
Suggested readings, activities in a library or similar: 10h.

Total hours of personal work of the students 92

Total volume of work: 150 hours

Prerequisites to follow this course are the following courses:
Linear and Multilinear Algebra, Topology of Euclidean spaces, Diferentiation of Real Functions of Several Variables.
Have studied or are studying Introduction to Ordinary Differential Equations