Mathematical Analysis II
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2006/2007 - 2S
Cycles of Study/Courses
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
Miguel Mendes; Curvas, Superfícies e Campos de Vectores
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
Robert C. Wrede; Schaum's outline of Advanced Calculus
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
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.
Observations
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Working time estimated out of classes: 4 hours