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Mathematics II

Code:

LTE12106

Sigla:

MATII

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2021/2022 - 2S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2122/course/view.php?id=310
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
LTE	64	Plano de Estudos	1	-	6	75	162

Língua de trabalho

Portuguese

Objetivos

The objective of curricular unit (UC) is to provide students with basic knowledge of linear algebra and skills to deal with the mechanisms of differential calculus in scalar and vector fields, mathematical tools of great importance in the professional training of a higher technician or engineer.

Resultados de aprendizagem e competências

In each topic, students must acquire the following skills:

1 - Matrices
a) Perform algebraic operations with matrices and understand the definition of the inverse of a matrix. Understand and apply the properties of the algebraic operations and of the inverse of a matrix.
b) Understand the notion and study the linear dependence and independence of the rows and columns of a matrix. Calculate the rank of a matrix, using elementary operations.
c) To solve and discuss a system of linear equations using the Gaussian elimination method. Find out if a matrix is invertible and calculate its inverse.

2 - Determinants
a) Understand the definition of determinant and its properties and apply the various methods to calculate a determinant.
b) Calculate the adjoint of a matrix, find out if a matrix is invertible and calculate its inverse using determinants. Use the Cramer's Rule.

3 - Eigenvalues and Eigenvectors
a) Understand the notions of eigenvalue and eigenvector of a matrix.
b) Calculate eigenvalues and eigenvectors of matrices.

4 - Vector Calculus
a) Understand the notions of inner product of vectors, norm and unit vector of a vector, calculate them and apply their properties.
b) Determine the angle between two vectors, the orthogonal projection and find out if a set of vectors is orthogonal or orthonormal.
c) Understand the notions of cross product and scalar triple product of vectors, calculate them and apply their properties.

5 - Differential Calculus in IRⁿ
a) Understand the notions of scalar and vector fields and study level curves and level surfaces.
b) Calculate limits and study the continuity of scalar and vector fields.
c) Understand the notions of partial derivative, differentiability, directional derivatives and gradient vector of scalar fields, understand their properties and calculate/study them. Determine the equations of the tangent plane and the normal line.
d) Study the differentiability, calculate the Jacobian matrix and directional derivatives of vector fields. Calculate the divergence and the curl operators.

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Knowledge acquired in the UC Mathematics I of the current curricular plan (Elements of Mathematics I and Elements of Mathematics II of the CTeSP are equivalent to Mathematics I).

Programa

1 - Matrices
a) Definition of matrix; algebraic operations with matrices; inverse of a matrix.
b) Linear dependence and independence of the rows and columns of a matrix, rank of a matrix and elementary operations.
c) Systems of linear equations; matrix inversion.

2 - Determinants
a) Definition of determinant; properties; computing methods.
b) Applications of determinants: computing the inverse matrix using the adjoint matrix; Cramer's rule.

3 - Eigenvalues and Eigenvectors
a) Definition and geometric interpretation of eigenvalue and eigenvector of a matrix.
b) Method for computing the eigenvalues and eigenvectors of a matrix.

4 - Vector Calculus
a) Inner product of vectors, norm and unit vector of a vector and their properties.
b) Angle between two vectors, orthogonal projection; orthogonal and orthonormal sets of vectors.
c) Cross product and scalar triple product of vectors; properties and applications.

5 - Differential Calculus in IRⁿ
a) Scalar and vector fields; level curves and level surfaces.
b) Limits and continuity of scalar and vector fields.
c) Partial derivatives, differentiability, directional derivative and gradient vector of scalar fields; equations of the tangent plane and the normal line.
d) Differentiability, Jacobian matrix and directional derivatives of vector fields; divergence and curl operators.

Bibliografia Obrigatória

Apontamentos e exercícios; elaborados por docentes do Departamento de Matemática (Notes and exercises made by teachers of the Mathematics Department - available on the Moodle page, only in Portuguese; students who do not read Portuguese should see the books listed below)

Bibliografia Complementar

Giraldes, E., Fernandes, V. H. e Smith, M. P. M.; Curso de Álgebra Linear e Geometria Analítica, McGraw-Hill, 1995. ISBN: 972-8298-02-1
Luz, C., Matos, A. e Nunes, S.; Álgebra Linear (Volume I), ESTSetúbal/IPS, 2002. ISBN: ISBN 972-8431-16-9
Azenha, A. e Jerónimo, M. A.; Cálculo Diferencial e Integral em R e Rn, McGraw-Hill, Portugal, 1995, ISBN 972-8298-03-X., McGraw-Hill, Portugal, 1995. ISBN: 972-8298-03-X
Luís, G., Ribeiro, C., S., Murteira Bento J. F.; Álgebra Linear, McGraw Hill, 1985. ISBN: 972-9241-05-8
Strang, G.; Linear algebra and its applications, Fort Worth: Saunders, 1988. ISBN: 0-15-551005-3 (Advised for students who do not read Portuguese)
Apostol, T.; Calculus, Vol. II, Blaisdell Publishing Company, Massachusetts, 1969. ISBN: 0-471-00008-6 (Advised for students who do not read Portuguese)
Laudesman, E. M. e Hestenes, M. R.; Linear Algebra for Mathematics, Science and Engineering, Prentice─Hall International, New Jersey, 1992. ISBN: 978-3-030-21323-7 (Advised for students who do not read Portuguese; not available at the ESTSetúbal Library)

Métodos de ensino e atividades de aprendizagem

Mathematics II has a teaching load of 5 hours per week, divided into 3 hours of theoretical-practical classes (TP) and 2 hours of practical-laboratory classes (PL).

In the theoretical-practical classes are presented the basic concepts of the different subjects of the syllabus and the proofs of the main results, followed by problems solving. In this type of classes students will acquire an overview of the themes and their interconnections. 

In PL classes students will solve under the guidance of the teacher a set of exercises, to gain a deeper understanding of the subjects.

The consolidation of knowledge by students will be based on reading the materials provided, the autonomous resolution of exercises using the study material recommended and the support of the teachers in their office hours.

All the information and specific materials of Matemática II will be available on its page in the Moodle platform.

To assess their knowledge, students will be given three multiple-choice formative tests on this platform.

Comments:

Daytime classes will be on-site.  

The classes in after-work hours will be remotely, on the Microsoft Teams platform, in the Mathematica II 21/22 team (in the specific channels for TP and PL classes). Students in confinement/isolation can request to temporarily attend these classes by sending an e-mail to ana.matos@estsetubal.ips.pt (TP classes) and maria.ribeiro@estsetubal.ips.pt (PL classes).

For other questions regarding these classes students should send an e-mail to ana.matos@estsetubal.ips.pt.

Tipo de avaliação

Evaluation with final exam

Componentes de Avaliação

Designation	Peso (%)
Exame	100,00
Total:	100,00

Componentes de Ocupação

Designation	Tempo (Horas)
Estudo autónomo	87,00
Frequência das aulas	75,00
Total:	162,00

Obtenção de frequência

Approval can be obtained either by continuous assessment or by exam assessment.

All assessments will be on-site, except if the IPS President or the Director of ESTSetúbal decide otherwise.

Continuous assessment
The continuous assessment is based on two tests.
Let NT1 and NT2 be the grades of the tests (rounded to tenths).
The final grade, CF, will be calculated by the formula CF=0.5xNT1+0.5xNT2.
The approval conditions are the following:     
- if CF is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 6.5;      
- if CF is greater than or equal to 18, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 17 values.

Retrieving a test  

To meet the approval conditions (final average greater than or equal to 10 and classification in both tests greater than or equal to 6.5 values), a student who has a classification greater than or equal to 8.0 in at least one of the tests may recover one and only one of the tests, on the same day and time of the first date of exam.  A student who has obtained less than 8.0 in one of the tests, has not taken it or has dropped out can only recover that test.
Recovering a test to improve a grade will not be allowed.
To recover a test, the student has to register, according to the terms and deadlines that will be indicated in due time.

Exam Assessment
Students who choose not to carry out the continuous assessment or fail do obtain approval on it may attend the regular exams.

Let E be the grade obtained in the exam (rounded to the units):
- if E is greater than or equal to 10 and less than 18, the student is approved with final grade E;
- if E is greater than or equal to 18, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 17 values.

Comment
The tests have a duration of two hours and the exams of two and a half hours.

Fórmula de cálculo da classificação final

Continuous assessment
Let NT1 and NT2 be the grades of the tests (rounded to tenths) and CF=0.5xNT1+0.5xNT2 (rounded to units):
- if CF is greater than or equal to 10 and less than 18, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 6.5;
- if CF is greater than or equal to 18, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 17 values.

Exam Assessment
Let E be the grade obtained in the exam (rounded to the units):
- if E is greater than or equal to 10 and less than 18, the student is approved with final grade E;
- if E is greater than or equal to 18, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 17 values.

Trabalho de estágio/projeto

Avaliação especial (TE, DA, ...)

Students covered by Article 246 of the IPS Student Performance Assessment Guidelines students must contact the teacher responsible by e-mail (ana.matos@estsetubal.ips.pt), until the second week of the semester, to present their relevant specificities.

Melhoria de classificação

According with the Article 11 of the IPS Student Performance Assessment Guidelines.

Observações

1. Assessment rules (tests and exams)
Enrolment for tests and exams is required up to one week before the date of the test/exam, on the Moodle page. 
In tests and exams is mandatory to present an official identification document. 
In the tests and exams, the student can have his own consultation form, on an A4 sheet, handwritten by the student himself. Other consultation forms will not be provided.
The use of calculators is not allowed in tests and exams.
The handling or displaying of mobile phones or any other means of remote communication during tests or exams is not allowed (mobile phones must be turned off).

2. The rules for remote assessments, if necessary, will be published in due time on the Moodle page.

3. In case of fraud suspicion or other circumstance that leads to the need to confirm the knowledge evidenced in the resolution of a test or exam, the student may be called to a session, in person, which will focus only on that knowledge. If the student does not attend this session, without due justification, the test or exam will be considered invalid.

4. According to the IPS rules, whenever there is a situation of proven fraud in an assessment test, the student will be subject to the application of the IPS Student Disciplinary Regulations.

5. The office hours of the teachers will take place on the TEAMS platform, in the channels of the teachers in the Matemática II 21/22 team. Its schedule and the link to this page will be available on the Moodle page. The student must previously inform the teacher of his attendance. The student may request face-to-face support, which is subject to prior appointment by e-mail, with at least 24 hours in advance, and the teacher confirmation.

6. All communication with the students is carried out exclusively through the institutional e-mail. It is up to the student to periodically consult his e-mail account in the IPS domain and use it to communicate with the teachers and IPS services.