Mathematics II

Code:

EBE0007

Acronym:

MAT2

Keywords
Classification	Keyword
OFICIAL	Basic Sciences

Instance: 2014/2015 - 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	85	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 ande the gradient. Geometry of level surfaces. Chain rule, implicit differentiation. 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

Denis Auroux; http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm, MIT, 2007
Lang, Serge; Calculus of Several Variables, 3rd edition. , Springer, 1987. ISBN: ISBN: 0-387-96405-3
Larson, Ron and Edwards, Bruce; Calculus, 9th edition, Cengace Learning, 2010, Cengace Learning, 2010, 2010. ISBN: 978-0-547-16702-2
Anton, Bivens, Davis; Calculus, Wiley, 2005. ISBN: 0-471-48273-0

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.

 

 

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

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

Calculation formula of final grade

Type of evaluation/Formula Evaluation: it consists on two mandatory written text (40%+60%). In the final written examination,  the student can repeat one of the tests, and only one. The approval for the course requires a final grade equal to or higher than 9.5 (out of 20).