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Mathematical Analysis II

Code: EEC0007     Acronym: AMAT2

Classification Keyword
OFICIAL Mathematics

Instance: 2011/2012 - 2S

Active? Yes
E-learning page: http://moodle.fe.up.pt/
Responsible unit: Department of Electrical and Computer Engineering
Course/CS Responsible: Master in Electrical and Computers Engineering

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
MIEEC 395 Syllabus (Transition) since 2010/2011 1 - 8 99 213
Syllabus 1 - 8 99 213

Teaching language



To develop techniques of differential and integral calculus.

This course unit aims to develop students’ skills on the manipulation of concepts of this course unit and to develop their independent and creative reasoning.

Learning Outcomes:
1) To correctly apply mathematical techniques included in the program.
2) To select the appropriate mathematical tools to solve problems.
3) To clearly display techniques involved in problem solving.
4) To analyse and criticise results obtained in problem solving.

CDIO Syllabus: 1.1; 2.4

Educational activities:
1) In theoretical and in theoretical-practical classes students should actively take part of the discussion by answering to questions and questioning the processes in the formulation and problem solving.
2) Individual resolution of exercises during theoretical-practical classes. Students should be capable of identifying the mathematical concepts involved and study the support material related to them and apply them on the resolution of exercises.
3) Resolution of self-evaluation tests.


1. Vector functions and real variable
Domain, graph. Continuity and derivatives. Curve and tangent vector.

2. Systems of linear differential equations: Laplace transform
3. Real functions and vector variable
Domain, graph, level set of real function of vector variable
Topological notions
Limits and continuity; Calculation rules
Rn curves
Partial derivatives; Directional derivatives; Derivative of a function
Relationship between derivability and continuity
Gradient vector; Geometrical interpretation
Normal line and tangent plan at point on the surface in R3
Higher order derivatives
Implicit derivation
Chain rule
Taylor’s formula

4. Functions defined in Rn: maximum and minimum
Critical points; Classification of critical points
Conditioned maximum and minimum; Lagrange multipliers

5. Vector functions of vector variable
Limits and continuity; Differentiability
Derivative of a function at a point; Jacob matrix
Inverse function theorem

6. Multiple integrals
Double and triple integrals
Change of variable in multiple integrals; Polar, cylindrical and spherical coordinates

7. Line, surface and volume integrals

Mandatory literature

Boyce, William E.; Elementary differential equations and boundary value problems. ISBN: 0-471-31999-6
M. do Rosário de Pinho e M. Margarida A. Ferreira; Apontamentos das aulas teóricas de AM2, 2007
Larson, Roland E.; Cálculo com geometria analítica. ISBN: 85-216-1108-0
S.K. Steib and A. Barcellos; Calculus and Analytic Geometry , McGraw Hill
Simmons; Calculus with Analytic Geometry, McGraw Hill.
Paula Rocha; Cálculo II, Universidade de Aveiro

Complementary Bibliography

Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3
Ana Breda e Joana N. Costa; Cálculo com funções de várias variáveis , McGraw Hill

Teaching methods and learning activities

Teaching Methods
Theoretical classes:
Presentation of problems; Discussion and deduction of results in the scope of this course unit
Students should prepare the theoretical classes by:
1) studying the recommended bibliography and themes covered in class;
2) trying to solve basic problems about the same themes;
3) taking notes of questions about that theme.

Theoretical-practical classes:
Every week students have to prepare the exercises given by the professors. In theoretical-practical classes the exercises will be discussed and questions regarding the exercises will be answered.

Self-evaluation tests:
During the semester, students will be encouraged to do some self-evaluation tests. The grade of these tests will not be taken into account in the final grade of this course unit. They aim to act as an orientation to the students, so that they know if they master the theoretical and practical concepts.

Office hours:
Every Wednesday from 2.30 pm to 4.pm in room B216, it will take place a session where students can ask questions and clarify their doubts.
Outside this time, professors will schedule an office hour to talk to students.


Physical sciences > Mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description Type Time (hours) Weight (%) End date
Attendance (estimated) Participação presencial 70,00
Exame 2,00
Teste 1, Teste 2 e Teste 3 Exame 3,50
Exercises Exame 35,50
Total: - 0,00

Amount of time allocated to each course unit

Description Type Time (hours) End date
Individual study Estudo autónomo 105
Total: 105,00

Eligibility for exams

To be admitted to exams, students cannot miss more classes than allowed by the rules (25% of theoretical-practical classes). They have also to attend to half of the self-evaluation tests (“quiz” on moodle).

Students, who attended this course in 2008/2009, do not need to attend classes this year. However, if students enrol in theoretical-practical classes, they will be assessed as if they were attending this course for the first time.

Calculation formula of final grade

Assessment components:
1) First test (T1) – 21st April
2) Second test (T2)- 9th June
3) Exam (E)
3) Recurso (resit) exam (R)

Final Grade will be based on the tests (T1 + T2, each of them will be classified from 0 to 10) OR on the final exam (E or R).

Students, who complete the course unit by attending the two tests, do not need to attend the final exam. They can improve their grades by attending the recurso (resit) exam.

Special assessment (TE, DA, ...)

Students with a special status (working students or military personnel) do not need to attend classes. They will be assessed based on the final exam (0 to 20).
Students, who completed Mathematical Analysis 2 in the previous syllabus, can opt to be assessed partially, by only attending the last three chapters of the program. They do not need to attend theoretical-practical classes. However, it is advisable that they attend to the classes, in which those themes will be covered. They will have to attend either the final exam and/or recurso (resit) exam.

Classification improvement

Students can improve their grades by attending the recurso (resit) exam. It is worth 100% of the final grade.


Final exam is a closed book exam and students are not allowed to use calculators. It will be given a form.
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