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Mathematics II

Code:

LEM12107

Sigla:

MAT II

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2023/2024 - 2S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2324/
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
EM	132	Plano de Estudos	1	-	6	75	162

Língua de trabalho

Portuguese

Objetivos

The goal of this curricular unit (UC) is to provide students basic knowledge of linear algebra and skills to deal with the mechanisms of differential calculus in scalar and vector fields, mathematical tools of significant 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) To perform algebraic operations with matrices and to understand the definition of the inverse of a matrix. To understand and apply the properties of the algebraic operations and of the inverse of a matrix.
b) To understand the notion and to study the linear dependence and independence of the rows and columns of a matrix. To calculate the rank of a matrix, using elementary operations.
c) To solve and discuss a system of linear equations using the Gaussian elimination method. To find out if a matrix is invertible and calculate its inverse.

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

3 - Eigenvalues and Eigenvectors
a) To understand the notions of eigenvalue and eigenvector of a matrix.
b) To calculate eigenvalues and eigenvectors of matrices.

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

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

Modo de trabalho

Presencial

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

Knowledge acquired in the CU Mathematics I of the current curricular plan.

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 versor 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
Magalhães, L. T.; Álgebra Linear como introdução à Matemática Aplicada, Texto Editora, 1989. ISBN: 972-47-0007-0
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 of the provided materials, autonomous resolution of exercises using the recommended study material, and the support of the teachers in their office hours.

All the information and specific materials of Mathematics 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.

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.

Continuous assessment
The continuous assessment is based on two tests.
Let NT1 and NT2 be the grades of the tests in a scale 0-20 (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 7.5.      
- 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.

Supplementary test opportunity
To meet the approval conditions (final average greater than or equal to 10 and classification in both tests greater than or equal to 7.5), a student who has a classification greater than or equal to 7.5 in at least one of the tests may repeat 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 7.5 in one of the tests, has not taken it or has dropped out can only repeat that test.
If a student takes a repetition test, the grade obtained in the repetition substitutes the grade originally obtained in the corresponding test.
Repetition of a test with the aim of improving an approval final grade will not be allowed.
To repeat a test, the student must register, according to the terms and deadlines that will be indicated in time.

Exam Assessment
Students who choose not to carry out the continuous assessment or fail to obtain approval on it may attend the regular exams.
Assessment by final exam follows the already mentioned rules, with the following conditions, where E is the grade obtained in the exam in a scale 0-20 (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.

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

Comment
The tests have a duration of two hours and the exams of two and a half hours. They consist of questions with open answer to be developed by the student and the grade in a scale 0-20.

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 7.5.
- 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.

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.

Provas e trabalhos especiais

Trabalho de estágio/projeto

Avaliação especial (TE, DA, ...)

Students covered by specific rights in terms of test and exam assessment, granted by of the IPS statutory normative, must contact the responsible teacher by e-mail (patricia.ribeiro@estsetubal.ips.pt), until the second week of the semester, to present their relevant specificities.

Melhoria de classificação

According with IPS statutory normative.

Observações

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, it is mandatory to present an official identification document. 

In tests and exams, students can have their own consultation A4-size paper sheet, handwritten by the student himself. Other kind of consultation materials will not be provided or allowed.

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 time on the Moodle page.

3. In case of fraud suspicion or other circumstance that leads to the need to assess 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. Timetable for the students tutorial support will be published in the UC's Moodle page. The student must previously inform the teacher, by e-mail, up to one hour from the tutorial start. 

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