Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 3 | 2 |
To assimilate the contents of the subject, it is convenient to have a previous knowledge of calculus in several variables (derivation, integration, implicit function theorem), differential equations (theorem of existence and uniqueness of solutions), algebra and linear geometry (diagonalization of self-adjoint endomorphisms, quadratic forms) and topology (homeomorphism, index of a flat curve, Euler characteristic).
The concepts and notions of differential geometry and vector calculus are basic for understanding the physical reality that surrounds us. Their technical applications are also important in the field of engineering, where the objects of study can be represented geometrically by non-linear elements of the three-dimensional space, that is, basically by curves and surfaces.
The main objective is to study the geometric notions that allow theoretical characterization of the shape of these elements (curvature and torsion in the case of a curve, first and second fundamental form in the case of a surface), as well as to develop methods to calculate their metric characteristics (length, area, etc.). It is also important to relate the invariants associated with a curve contained on a surface with the notions and magnitudes of the latter. All these topics will be dealt with in the first two blocks of the course.
In the third block, the classic notions of the vector calculus will be introduced: vector fields and their line and surface integrals, as well as the integral theorems of Green, Gauss and Stokes that relate them. These results will be obtained as particular cases of the Stokes theorem for differential forms. Numerous applications of these theorems will be presented, both in physics and geometry.
1. Curves
Parameterization and length.
Curvature and torsion. Frenet Formulas
Osculating sphere.
2. Surfaces.
Regular surfaces.
First fundamental form.
The Gauss map and the second fundamental form.
Normal curvature and geodesic curvature.
Intrinsic geometry of surfaces. Gauss Egregium theorem.
Geodesics
The default theorem.
3. Vector calculus.
Differential forms.
Varieties and varieties with boundary.
Integration
Stokes theorem and applications.
The Gauss-Bonnet theorem.
The course consists of:
1. Theory classes, where the fundamental concepts, illustrated with abundant examples, will be introduced, and the program's themes will be explained. Students will be encouraged to actively participate in class.
2. Problem solving classes, where significant exercises will be solved and certain questions that clarify the notions introduced into the theory classes will be analyzed in detail. The heuristic methods of solving mathematical problems that are characteristic of the subject will be emphasized.
3. Seminars, where the students, under the guidance of the professor, will develop and deepen certain topics that complement the subjects explained in the theoretical classes.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Problem solving classes | 30 | 1.2 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
Theory classes | 45 | 1.8 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
Tutoring | 15 | 0.6 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
Type: Supervised | |||
Seminars | 28 | 1.12 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
Type: Autonomous | |||
Autonomous study | 160 | 6.4 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
The mark of the continuous evaluation of the subject, AC, will be obtained from:
1. the marks of two partial exams, E1 and E2,
2. the mark of the delivery of exercises (from the classes of problems or seminars), P,
according to the formula: AC = 0.30 E1 + 0.45 E2 + 0.25 P.
The student passes the subject if AC is greater than or equal to 5. Otherwise, the student has a recovery examination giving rise to a mark ER. The recovery examination can be used to improve the grade but notice that, in all the cases, ER will replace the sum of the two partial examination marks, E1 + E2 . The mark P of delivery of exercises is NOT recoverable and therefore the mark ER of the recovery examination will have a weight of 75% in the final grade. To be able to attend the recovery exam, the student must have been previously evaluated of continuous assessment activities that are equivalent to 2/3 of the total.
It is considered that the student presents himself for the evaluation of the course if he has participated in evaluation activities that exceed 50% of the total.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Delivery of solved problems P | 25% | 10 | 0.4 | 2, 3, 12, 10, 9, 8, 7, 5, 6, 11, 4, 13 |
Exam E1 | 30% | 4 | 0.16 | 2, 3, 12, 10, 9, 7, 5, 6, 11, 4 |
Exam E2 | 45% | 4 | 0.16 | 2, 3, 12, 10, 9, 7, 5, 6, 11, 4 |
Recovery exam ER | 75% | 4 | 0.16 | 3, 1, 12, 10, 9, 7, 5, 6, 11, 4 |
1. Manfredo P. do Carmo. Geometría diferencial de curvas y superfícies. Alianza Editorial. 1990
3. Sebastián Montiel y Antonio Ros. Curvas y superfícies. Proyecto Sur. 1998
4. Michael Spivak. Cálculo en Variedades. Ed. Reverté. 1970
5. Joan Girbau. Geometria diferencial i relativitat. Manuals de la UAB. 1993
eBooks:
2. Theodore Shifrin. Differential Geometry: A First Course in Curves and Surfaces. 2021
6. Victor A. Toponogov. Differential Geometry of Curves and Surfaces. Birkhäuser. 2006
7. Shoshichi Kobayashi. Differential Geometry of Curves and Surfaces. Springer. 2019
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