Real Analysis III

Code:

M2010

Acronym:

M2010

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	1	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	1	Official Study Plan	2	-	6	56	162
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:M	114	Official Study Plan	2	-	6	56	162
L:Q	0	study plan from 2016/17	3	-	6	56	162

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

Program

(Main topics)

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 rotational. 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

Explanation of the several theoretic topics on the blackboard . Strong interconnection between the theoretical and practical classes at the level of exercises and examples and of theoretical results.

As there are only two hours/week for theorectical presentation  it is assumed that the student uses the compulsory bibliography to complement the exposition made in the theoretical class, to explore the examples and to solve exercises. This must be done in an autonomous way but under general orientation given by the professor

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	108,00
Frequência das aulas	54,00
Total:	162,00

Eligibility for exams

Not applicable

Calculation formula of final grade

• 
  

  Evaluation in the first season is performed through two mandatory tests, T1 and T2, each one worth 10 points with a duration of 2 hours. The first test is done during the semester and the second during the first season of exams.

  
  


• 
  

  The final mark will be the sum of the marks obtained in each test. Approval requires a sum not inferior to 9,5 points. Those obtaining a grade higher than 17 in the exam or in the full collection of tests must do an extra short written exam to confirm their mark.

  
  


• 
  

  The second season exam consists of two independent tests, each one worth 10 points with a duration of 2 hours. In any situation (seeking approval or improving their grade), in the second season the student may choose to do both parts or just one of them, R1 and R2, R1 or R2. They must transmit that intention to the professors before doing the exam. The final grade, CF, is the sum of the maximum grades obtained in each part of the evaluation: CF=max{T1, R1}+max{T2,R2}. Approval requires a sum not inferior to 9,5 points. Those obtaining a grade higher than 17 in the exam or in the full collection of tests must do an extra short written exam to confirm their mark.

  
  


• 
  

  All other evaluation situations unforeseen in the previous points, in particular improving grades from previous seasons and substitution exams previewed in the regulations, will be performed through a unique exam, not exceeding 3 hours of duration, which may be  preceded by a simple oral exam to verify if the student is minimally prepared to realize the exam.

  
  

 

 

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.