Unit GEOMETRY IV
- Course
- Mathematics
- Study-unit Code
- 55099109
- Curriculum
- In all curricula
- Teacher
- Nicola Ciccoli
- Teachers
-
- Nicola Ciccoli
- Hours
- 68 ore - Nicola Ciccoli
- CFU
- 9
- Course Regulation
- Coorte 2020
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica
- Academic discipline
- MAT/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Differentiable curves in R^n and embedded surfaces in R^3.
Elements of topology. - Reference texts
- A. LOI, Introduzione alla Topologia Generale, Aracne 2013.
M. ABATE, F. TOVENA, Curve e superfici, Springer, 2006.
I. SINGER, A THORPE, Lecture Notes on Elementary Topology and Geometry, Undergraduate Texts in Math, Springer.
M. P. DO CARMO, Differential Geometry of curves and surfaces, Pearson, 1976.
G. CAMPANELLA
Curve e superfici - Esercizi svolti, Aracne, 2006.
More texts in English may be signalled upon request.
These texts are signalled as general reference text, meaning that we will not follow closely anyone of these. The students are required to find their own paths through them, according to a process of individual learning that cannot be predetermined by the professor.
Preparing the exam on the basis of personal notes only is strongly discouraged.
I am available to help students, during office hours(in person or online), in finding more precise bibliographical reference (through the wide variety that can be find at the Library or on the Web). - Educational objectives
- Being capable of computing the main geometric invariants of curves and surfaces, and reconstructing parametric equations of curves and surfaces satisfying suitable conditions.
Being able to indipendently read some mathematical literature, finding their own path through the teaching material. - Prerequisites
- It is unlikely that the students can profit from following this course without a solid background in multivariable calculus (comprehensive of existence and uniqueness theorems for ordinary differential equations) and linear algebra. A good background in affine geometry, elementary metric topology and some general basic algebraic concepts would be of help.
- Teaching methods
- Lectures covering all topics of the course.
Exercises similar to those contained in the written exam. It has to be clarifies that not all exercises rely on a "standard" and some of them have the explicit purpose to stimulate the student in applying his knowledge in not predetermined contexts. Tutoring sessions. The students are encouraged in being active parts of the learning process, trying to avoid memorization of computational routines. They are thus required a proactive approach towards the teaching material. In particular: not all proofs will be complete and some of the material will be only sketched leaving to the student to fill in details. - Other information
- In accordance with students there will be some organized study group activities (non mandatory).
Following lectures is not compulsory but strongly advised.
Office hours will be determined at the start of lectures. It will be possible to fix online office hours by fixing them via e-mail. - Learning verification modality
- The exam is composed of a written and an oral examination.
The written examination requires solving a non fixed number of exercises. It lasts 180 minutes. Its purpose is to verify the student ability to apply the theoretical content of the course in explicit examples. A minimum of 15 out of 30 points is required to be admitted to the oral examination. It is not allowed the use of electronical devices.
The oral examination, usually less than 45 minutes, is meant to verify how the student is mastering a correct exposition, his ability in reproducing proofs, his ability to connect materials between different arguments.
It is not possible to establish a fixed weight of the two evaluations in composing the final evaluation score.
There will be a mid-term exam. More details on its characteristic - it will certainly require a variety of different kind of verifications - will be given during the first lecture.
More informations on support services for disabled studentes and/or students with special needs can be found at http://www.unipg.it/disabilita-e-dsa - Extended program
- Basic topology theory as needed in the course: topological spaces and continuous maps.
Local theory of differentiable parametrized curves. Arclength, Frènet's frame, curvature and torsion. Reconstrcution problems. Hints on global problems.
Locale theory of parametrized surfaces in R^3. Differentiable functions, tangent space. First and second fundamental forms. Curvatures: principal, normal, mean, Gaussian.
Gauss' Theorema Egregium. Christoffel symbols, geodesics and geodetic curvature. Gauss-Bonnet theorem and global properties of surfaces.