Real Analysis III

Code:

M2033

Acronym:

M2033

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	0	Official Study Plan	2	-	9	72	243
L:MA	0	Official Study Plan	2	-	9	72	243

Teaching language

Portuguese

Objectives

To introduce, in a concrete way, the main results of Classical Analysis of several variables as well as the ones of Vector Analysis, emphasizing techniques specific to this area as well as their applications.

Learning outcomes and competences

The student is expected to learn the basic ideas and main results of the subject as well as becoming familiar with the main tools referred to in the syllabus.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Real Analysis I & II, Linear Algebra and Analytic Geometry I & II.

Program

1) Metric spaces. basic definitions and properties. Convergente and Cauchy sequencies. Complete metric spaces. Compact metric spaces. Connected and path-connected metric spaces.
2) Lipchitz maps and contracting maps. Banach's Theorem.
Functional spaces. Pointwise and uniform convergence.
3) The inverse and implicit function theorems.
4) Applications: Lagrange multipliers.
5) Parametrized curves. Velocity and aceleration. Curvature and torsion.
6) Paths and line integrals. Conservative, gradient and closed vector fields. Green's Theorem
7) Parametrized and regular surfaces in R^3. Tangent and normal spaces. Area and integral of scalar functions. Oriented surfaces.
8) Flow of a vector field. The gradient and the curl. Stokes and Gauss Theorems. Gauss' law.

Mandatory literature

Marsden, Tromba; Vector Calculus, W. H. Freeman and Company, 1988. ISBN: 0-7167-1856-1
Marsden, Hoffman; Elementary Classical Analysis, W. H. Freeman and Company, 1993. ISBN: 0-7167-2105-8
Elon Lages Lima; Espaços Métricos, Projecto Euclides, 2003. ISBN: 85-244-0158-3

Complementary Bibliography

Munkres James R.; Analysis on manifolds. ISBN: 0-201-51035-9
Serge Lang; Calculus of Several Variables, Springer, 1987. ISBN: 0-387-96405-3

Teaching methods and learning activities

Expository lectures. Interconnection between lectures and tutorial classes at the level of exercises and examples and of theoretical results.

It is assumed that the student uses the compulsory bibliography to complement the exposition made in the lectures, to explore the examples and to solve exercises. This must be done in an autonomous way but under general guidance given by the professor.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

Not applicable

Calculation formula of final grade

Evaluation in the first season is performed through a mandatory written examination. Approval requires mark not inferior to 9,5 points. 

All other assesment situations, in particular improving grades from other calls and substitution exams allowed by the regulations, will be performed through an oral or written exam which may be preceded by a simple oral exam to verify if the student is minimally prepared to take the exam.

Special assessment (TE, DA, ...)

Special exams will consist of an oral or written test, which may be preceded by a simple oral exam to verify if the student is minimally prepared to realize the exam.

Classification improvement

By exam in accordance with the regulations. Results from distributed assesment in previous academic years cannot be used.