GEOMETRY 2
Linear algebra. Basic notions of general topology. Calculus for functions in many variables
In accordance with the objectives of the course, the verification of the learning will be carried out through:
- a written test, in which the student must show that he has acquired the ability to verify the main properties of curves and surfaces on concrete examples, chosen from the list assigned during the course. The writing will last two hours and will consist of two problems. The written test mark will have a maximum score of 16. If you get at least 8 you can access the oral exam.
- an oral exam, during which the student must show that he has acquired the main notions and the proofs of the most relevant theorems of the course. The maximum oral mark is 16. The final mark is given by the sum of the written and oral marks. A grade greater than 30 automatically brings the “lode”.
EDUCATIONAL GOALS
At the end of the course, the student will be able to:
1. understand the fundamental notions of the theory of parametrized curves in Euclidean spaces.
2. illustrate both local and global aspects of their extrinsic geometry.
3. develop invariants of curves independent of the parametrization.
4. understand the theory of integration on surfaces, presenting some geometric applications
5. develop, through exercise on concrete examples, the ability to calculate the main quantities that describe the local geometry of a surface, such as the Gaussian curvature.
6. Calculate global topological invariants such as the Euler characteristic
The course is divided into two parts: the first is dedicated to differentiable curves in the plane and in space and provides an overview of concepts and results that we will try to extend, in the second part, to the differentiable surfaces of space.
A) Differentiable curves
1) Smooth curves in space and their length. 2) Regular curves, tangent line and arc parameter.
3) Curves in space, Frenet trihedron, Curvature with sign and Frenet formulas.
4) Fundamental theorems of the global geometry of plane curves.
B) Abstract differentiable manifolds immersed in Euclidean spaces.
1) Regular surfaces of the Euclidean space.
2) Level surfaces.
3) Smooth functions between regular surfaces.
4) Tangent and differential plane of a smooth map.
5) Vector fields on surfaces, their integral curves
5) First fundamental form and isometries.
6) Orientability and Gauss map.
7) Second fundamental form and curvatures.
8) Orientability and integration on compact surfaces.
9) Parallelism and geodesics.
9) Local Gauss Bonnet theorem.
10) Triangulations and global Gauss Bonnet theorem for orientable compact surfaces.
The lectures and exercises will be conducted by the teacher using a tablet, and the notes of each lecture or exercise will be available on e-learning at the end of each week. A list of problems to be done at home will be assigned, with suggestions and tracks provided during the exercises in class.
email:
riccardo.re AT uninsubria.it
Contact the teacher via email for any clarification