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Complements of Mathematics

Code: EIC0009     Acronym: CMAT

Classification Keyword
OFICIAL Mathematics

Instance: 2018/2019 - 2S Ícone do Moodle

Active? Yes
Responsible unit: Mathematics Section
Course/CS Responsible: Master in Informatics and Computing Engineering

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
MIEIC 200 Syllabus since 2009/2010 1 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
José Augusto Trigo Barbosa
Maria Luisa Romariz Madureira

Teaching - Hours

Lectures: 2,00
Recitations: 2,00
Type Teacher Classes Hour
Lectures Totals 2 4,00
Maria Luisa Romariz Madureira 1,00
José Augusto Trigo Barbosa 3,00
Recitations Totals 8 16,00
Alexandre Miguel Prior Afonso 8,00
Maria Luisa Romariz Madureira 1,00
Ana Maria Azevedo Neves 4,00
José Augusto Trigo Barbosa 3,00
Mais informaçõesLast updated on 2019-02-06.

Fields changed: Calculation formula of final grade, Bibliografia Complementar, Bibliografia Obrigatória, Componentes de Avaliação e Ocupação

Teaching language

Suitable for English-speaking students


1- BACKGROUND The main aim is to introduce fundamental mathematical concepts by developing the ability to analyze problems and results and also to acquire mathematical precision. These aspects form an educational background for other subjects in the curricula.

2- SPECIFIC AIMS Enhance the students reasoning capacity and knowledge of essential mathematical concepts. The students should acquire solid theoretical and practical training on the main concepts and results of differential and integral calculus of several variables, including the basic theorems of calculus.

 3- PREVIOUS KNOWLEDGE Functions and graphs. Differential and integral calculus in R1. Vector algebra. Lines and planes in R3.

4- PERCENTAGE DISTRIBUTION Scientific component:75% Technological component:25%

5- LEARNING OUTCOMES Knowledge and understanding: Partial and directional derivatives for real-valued and vector-valued functions; gradient vector. The chain rule for real-valued and vector-valued functions including implicit functions. Apply parametric curves and surfaces in R3 to calculate line and surface integrals. Establishment of the relationship between the line integral and the surface integral based on the Green’s, Stokes and Gauss Theorems.


Learning outcomes and competences

Students should deepen their knowledge on the concepts of line, surface, double and triple, integrals and their applications and improve the knowledge of vector and scalar fields.  Students should be able to apply these concepts in engineering problems

Working method



1- VECTOR FUNCTIONS. Properties. Curves. Arc length. Curvature. 2- FUNCTIONS OF SEVERAL VARIABLES. Quadric surfaces. Level curves and level surfaces. Partial derivatives. Limits and continuity. 3-GRADIENTS. Differentiability and gradient. Gradients and directional derivatives. The mean-value theorem. Chain rules. Maximum and minimum values. Differentials. 4- DOUBLE AND TRIPLE INTEGRALS. The double integral over a region. Evaluating double integrals using polar coordinates. Triple integrals. Cylindrical coordinates. Spherical coordinates.  Jacobians; changing variables in multiple integration. 5- LINE INTEGRALS AND SURFACE INTEGRALS. Line integrals. Properties. Line integrals respect to arc length. Green’s theorem. Parametrized surfaces. Surface área. Surface integrals. Divergence and curl. The divergence theorem. Stokes’s theorem

Mandatory literature

José Augusto Trigo Barbosa; Noções sobre Geometria Analítica e Análise Matemática, Efeitos Gráficos, 2018. ISBN: 978-989-99559-7-4
José Augusto Trigo Barbosa; Noções sobre Análise Matemática, Efeitos Gráficos, 2019. ISBN: 978-989-54350-0-5
José Augusto Trigo Barbosa; Apontamentos de apoio às aulas teóricas, 2017

Complementary Bibliography

Madureira Maria Luísa Romariz Universidade do Porto. Faculdade de Engenharia; Problemas de integrais de linha e superfície e de séries de Fourier. ISBN: 978-989-99559-2-9

Teaching methods and learning activities

Theoretical classes will be based on the presentation of the themes of the course unit. These classes are aimed to motivate students, where examples of application will be showed. Theoretical-practical classes will be based on the analysis and on problem solving by students, where they have to apply tools and mathematical concepts taught in theoretical classes. These classes are aimed to assess students’ understanding and dexterity of the themes of the course unit.


Physical sciences > Mathematics > Mathematical analysis > Functional analysis

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation Weight (%)
Participação presencial 0,00
Teste 100,00
Total: 100,00

Amount of time allocated to each course unit

Designation Time (hours)
Estudo autónomo 106,00
Frequência das aulas 56,00
Total: 162,00

Eligibility for exams

Students cannot miss more classes than allowed in the regulation. Exception for working students (article 8 of General Evaluation Rules of FEUP).

Calculation formula of final grade

The student must attend to two written exams, with the duration of 2 hours each. Each exam comprises two different parts: a theoretical part, which worth 15% of the final mark and a theoretical-practical part which worth 85% of the final mark. Final mark will be based on the average grade of the two exams.

Exams are scheduled for these dates: 1st written exam: 24 April 2019; 2nd written exam: 06 June 2019.

In order to pass the course the studente must: i) satisfy what is arranged in Article 4 of General Evaluation Rules of FEUP; ii) earn a grade of five point five out of twenty or better in each of the exams.

At the end of the semester students will be able to attend a new exam in order to improve their final grade. This exam may either only test a part of the program, or the whole program. Only students who got admission to exam can attend this exam.

Students who have passed the end of the evaluation process
Distributed and intend to carry out. to improve the classification obtained, one of the revaluation evidence (in appeal) should make their registration in Academic Services of FEUP.

The achievement of a classification of 20 values requires an oral test.

Date of the exam: 26 June 2019.

Special assessment (TE, DA, ...)

According to Articles 10 and 14 of General Evaluation Rules of FEUP.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to Article 11 of General Evaluation Rules of FEUP.

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