Differential Geometry

Code:

M355

Acronym:

M355

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 2S

Active?	Yes
Web Page:	http://elearning2.fc.up.pt/aulasweb0910/course/view.php?id=2289
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	1	Plano de Estudos a partir de 2008	3	-	7,5	-
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:M	14	Plano de estudos a partir de 2009	1	-	7,5	-
			2
			3
M:AST	0	Plano de Estudos do Mestrado em Astronomia	1	-	7,5	-

Teaching language

Portuguese

Objectives

Students should acquire knowledge about the application of methods of differential and integral calculus to the study of geometry with emphasis on the differential geometry of surfaces. They should be able to apply this knowledge independently to analyze and solve mathematical problems in contexts where methods of differential geometry are relevant.

Program

Basic point set topology necessary for the development of the theory. Smooth surfaces and implicitly defined surfaces. Differentiable functions on surfaces and maps between surfaces. Tangent plane. Orientation. First fundamental form. Area. Geodesics and normal curvature. Geodesic curves. The second fundamental form. Curvature of surfaces. Gauss' Theorema Egregium. The Gauss-Bonnet theorem.

Additionally, one or more of the following optional topics will be treated:

- Global differential geometry of curves and surfaces;
- Classification of topological surfaces;
- Surfaces and abstract manifolds;
- Surfaces of constant curvature and the parallel axiom;
- Differential forms and applications;
- Variational problems and minimal surfaces;
- Applications of differential geometry to physics.

Mandatory literature

Manfredo P. do Carmo; Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

Complementary Bibliography

Paulo Ventura Araújo; Geometria Diferencial, IMPA, 2004

Teaching methods and learning activities

Classes: Lectures, problem sessions, student presentations.

Homework: regular homework will be set, submission is mandatory for eligibility for assessment.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	54,00
First test	Exame	3,00		2012-04-17
Second test	Exame	3,00		2012-06-05
	Total:	-	0,00

Eligibility for exams

The following requirements are mandatory for eligibility for assesment:

- Participation in 75% of classes (T and TP separately);
- Submission of 75% of homework sets.

Calculation formula of final grade

The distributed assessment consists of two tests. The final mark is calculated as the average of the individual marks of the tests. The second test contains an optional part, corresponding to the material assessed in the first test. If the student chooses to hand in this optional part, the corresponding mark substitutes the mark of the first test in the calculation of the final mark.

Students who are eligible for assessment and have not passed the course in the distributed assessment may take a resit exam in the corresponding exam period, the final mark being the one obtained in this exam.

To obtain a mark higher than eighteen, the student may be required to complete complementary assessment. In order to access the complementary assessment, the student must have a mark of at least eighteen in the regular assessment.

Special assessment (TE, DA, ...)

By special exam, oral or written.

Classification improvement

The final mark obtained by distributed assessment can be improved by taking the resit exam in one of the two exam periods immediately following the course, in accordance with the "Regulamento de Avaliação do Aproveitamento dos Estudantes" of FCUP.

Observations

Remarks
Pre - requisites:

Fundamentals of Linear Algebra, Calculus, Geometry and Analysis in IRn. (Vector Analysis / Analysis)