Mathematics II

Code:

EBE0007

Acronym:

MAT2

Keywords
Classification	Keyword
OFICIAL	Basic Sciences

Instance: 2018/2019 - 2S

Active?	Yes
Responsible unit:	Population Studies
Course/CS Responsible:	Master in Bioengineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIB	86	Syllabus	1	-	6	56	162

Teaching language

Portuguese

Objectives

This course unit aims to familiarise students with mathematical concepts and tools to study functions of various variables, as well as to develop their analytical reasoning.

Learning outcomes and competences

The student is expected to learn the basic ideas and main results of the subjects referred  in the syllabus, as well as becoming familiar with the main tools  of the classical Real analysis and Vector Analysis. In particular s/he must develop capacity at the level of differential and integral calculus to several variables.

Working method

Presencial

Program

The euclidean space. the Inner and cross products. Norm and distance.

Curves. Velocity and speed, length, normal and tangencial accelerations, curvature.

Some metric notions. Functions of several variables. Partial derivatives and the gradient. Geometry of level surfaces. Chain rule. Hight order parcial derivatives.

Taylor´s polynomial, remainder's formula. (1 or 2 variables).

Maximun and minimum: Hessian matrix and Lagrange multipliers.

The Riemann integral and Fubini´s Theorem. Multiple integrals; polar, cylindric and spherical coordinates. Change of variables and the Jacobian Matrix.

Paths, line integrals and gradient vector fields. Green's Theorem; applications.

Parametrized surfaces: geometry, area and surface integrals, orientation, induced orientation in the boundary of the surface. Flux over a surface.

The divergence and rotational operators. Stokes' and Gauss Theorems. Applications. Faraday's and Gauss laws.

Mandatory literature

Tromba, Anthony; Marsden Jerrold; Vector Calculus, 3rd edition, W. H. Freeman and Company, 1988. ISBN: ISBN 0-7167-1856-1
Stewart, James; Multivariable Calculus, 7th edition, Cengace Learning, 2012. ISBN: 978-0-538-49787-9

Complementary Bibliography

Anton, Bivens, Davis; Calculus, Wiley, 2005. ISBN: 0-471-48273-0
Lang, Serge; Calculus of Several Variables, 3rd edition. , Springer, 1987. ISBN: ISBN: 0-387-96405-3

Teaching methods and learning activities

The contents of the syllabus are introduced in theoretical and intuitive presentations, where a lot of examples and applications are given and discussed to illustrate the concepts and results. There are also practical lessons, where exercises and related problems, previously suggested, are solved. All resources are available for students at the unit’s web page.
Due to the aims of the course most of the results are presented with formal statement, sufficiently motivated and supported in geometric and analytical arguments. Most proofs are omitted but are available in the required bibliography.

 

 

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	106,00
Frequência das aulas	52,00
Total:	158,00

Eligibility for exams

To be admitted to exams, students have to attend the tests

Calculation formula of final grade

The syllabus will be divided into two mandatory tests, the first one worth 8 points and the second one  worth 12 points (each of 2 hours long each).

 

The first test will be held in the mid of the semester, the second test will be held during the first season exam period. The final mark of the first season is the sum of the marks obtained in each test (equal or above 9,5).

 

Second season exam:

- In the second season exam, students may repeat both tests or just one of them (except when they are just trying to improve their mark upon previous approval).

- The mark of each part is the maximum of the marks obtained in the respective tests in both seasons (except when they are just trying to improve their mark upon previous approval in 2017/18).

- In cases of improvement of the classification the students approved in 2017/18 or they choose the system of compulsory evaluation during the semester and follow the general rules or they choose to perform only the second season  examination and in this case they bhave to solve both parts of the exam.

- The final mark of the second season is the sum of the marks obtained in both parts, rounded to integers.

 

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Any other extraordinary evaluation may be preceded of an oral exam.