Mathematical Analysis II

Code:

EC0006

Acronym:

AMAT2

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2008/2009 - 2S

Active?	Yes
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	404	Syllabus since 2006/2007	1	-	7	75	187

Teaching language

Portuguese

Objectives

To introduce foundamental concepts in view of the analyse of functions of several variables. To develop the ability to analyse problems and results and aquire mathematical precision. To induce an educational background for other subjects in the curricula.

Program

CHAPTER 1 – BRIEF NOTIONS OF TOPOLOGY IN IRn .

CHAPTER 2 – INFINITESIMAL ANALYSIS
The geometry of scalar functions and vector fields.
Continuity and Differentiability.
Taylor series of a scalar function.
Local and global maxima and minima of scalar functions. Constrained extrema.
Implicit function theorem and Inverse function theorrem.
CHAPTER 3 – DIFFERENTIAL GEOMETRY AND INTEGRATION
Curves.
Surfaces.
Integration of scalar functions in IRn: areas and volumes of regions with nonempty interior in IR2 e IR3 .
Integration of scalar functions in curves and surfaces.
Integration of vector-valued functions in curves and surfaces.

CHAPTER 4 – VECTOR FIELDS AND DYNAMICS
Vector Fields and its associated Flow.
Equilibria and their stability.
Conservative Fields and Hamiltonians: potencial function vs energy.
CHAPTER 5 – CLASSICAL THEOREMS OF ANALYSIS AND APPLICATIONS

Mandatory literature

Maria do Carmo; Lições de Análise Matemática 2

Complementary Bibliography

Azenha, Acilina; Elementos de cálculo diferencial e integral em IR e IRn. ISBN: 972-8298-03-X
Lima, Elon Lages; Curso de análise. ISBN: 85-244-0047-1
Marsden, Jerrold E.; Vector calculus. ISBN: 0-7167-0462-5

Teaching methods and learning activities

It is essentially a formative subject, coordinating fundamental theoretical knowledge with some approaches which are necessary in the subjects placed ahead in the curricula. At this level it is important to develop intuitive concepts as well as computational ability. The concepts are exposed in a clear and objective way, making frequent use of examples of physical or geometrical nature. The use of the software Maple is incentivated, as a working tool, namely through the execution of a practical project.

Evaluation Type

Evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	65,00
	Total:	-	0,00

Eligibility for exams

Limitations are as fixed by the school regulations (Artº 4-nº1).

Calculation formula of final grade

The note of the final exame.

Special assessment (TE, DA, ...)

Written examination.
SPECIAL RULES FOR MOBILITY STUDENTS:
Proficiency in Portuguese and/or English;
Previous attendance of introductory graduate courses in the scientific field addressed in this module;
Evaluation by exam and/or coursework(s) defined in accordance with student profile.

Observations

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Working time estimated out of classes: 4 hours