|
Apllied Mathematics
Scholar Year: 2020/2021 - 1S
| Code: |
EM21212 |
|
Acronym: |
MA |
| Scientific Fields: |
Ciências Base |
Courses
| Acronym |
N. of students |
Study plan |
Curricular year |
ECTS |
Contact time |
Total Time |
| EM |
134 |
|
2º |
6,0 |
60 |
162,0 |
Teaching language
Portuguese
Intended learning outcomes (Knowledges, skills and competencies to be developed by the students)
1. Multiple integrals
Identify and geometrically represent the integration region.
Recognize the most effective integration order.
When necessary, make the most appropriate change of variable.
Calculate areas, volumes, masses, centers of mass and moments of inertia using multiple integrals.
2. Ordinary Differential Equations (ODE)
Recognize and solve the following five types of 1st order ODE: separable variable ODE; homogeneous ODE; linear ODE; Bernoulli EDO and exact ODE.
Identify the characteristic polynomial of an nth order linear ODE with constant coefficients and determine and classify the roots of this polynomial.
Determine the general solution of an nth order homogeneous linear ODE with constant coefficients.
Determine the general solution of an nth order linear ODE with constant coefficients, not homogeneous, in certain specific cases.
3. Laplace transforms
Calculate the Laplace transform of a function using the definition.
Determine, using the properties, direct and inverse Laplace transforms. Solve ODE with Laplace transforms.
4. Series
4.1. Numerical Series
Identify the following series, determine their nature and calculate the sum, when convergent: geometric series; arithmetic series; telescopic series.
Identify a series of non-negative terms and apply the most appropriate of the following tests to determine their nature: Integral Test; 1st and 2nd Comparison Tests; Reason Test; nth root Test.
Identify alternating series and study their nature (simple/absolute convergence or divergence).
4.2. Power Series
Determine the convergence range of a power series.
Represent functions in power series of (x-a), when necessary using the Derivation and Primitive Power Series Theorem.
5. Fourier series
Determine the Fourier coefficients and the Fourier series of a periodic function.
Geometrically interpret periodic functions, even/odd periodic functions and perform even/odd periodic extensions.
Syllabus
1. Multiple integrals
Double and triple integrals: definition, properties and applications. Change of variables in double and triple integrals.
2. Ordinary Differential Equations (ODE)
Definitions and examples. First order ODE: solving methods and techniques for separable variables equations; homogeneous equations; linear equations; Bernoulli equations and exact equations. Nth order linear equations with constant coefficients: properties and solving methods.
3. Laplace transforms
Laplace transform and inverse Laplace transform: definitions and properties. Solving ODE with Laplace transforms.
4. Series
Numerical series: convergent series, properties and convergence tests. Absolute convergence.
Power series: domain of convergence; differentiation and integration of power series. Taylor series. Representation of functions in power series.
5. Fourier series
Fourier series of a periodic function. Computation of the Fourier coefficients. Fourier series of some usual functions. Properties of Fourier series. Convergence of Fourier series.
Teaching methodologies
Matemática Aplicada has a teaching load of 4h/week of theoretical-practical classes, in which the fundamental concepts are presented, proved some results and solved exercises that illustrate each topic.
In these classes students should acquire an overview of the themes and their interconnections, learn the correct and objective formulation of mathematical definitions, the precise enunciation of propositions and practice the deductive reasoning, as well as learn some applications to engineering of the various notions presented.
It will be up to the student, after each class, to conduct an autonomous study on the topics presented and deepen their knowledge, using the study material recommended, the bibliography and the support of the teachers in their office hours.
All the information and specific materials of Matemática Aplicada will be available on its page in the Moodle platform.
Classes will take place at distance, on the TEAMS platform, on the channel of each class in the Matemática Aplicada 2020/2021 team.
The consolidation of knowledge by students will be based on reading the materials provided and autonomous resolution of exercises.
Assessment methodologies and evidences
The student can obtain approval by continuous assessment and, should he fail, by exam assessment. The student can also choose to do only the exam assessment.
Test and exam assessments will be in person, except if the Government or the IPS President decide otherwise (due to the pandemic). In such case, the assessments will be carried out online and the evaluation standards and procedures will be updated and published on the Moodle page.
WARNING: Due to the Government measures to control the COVID 19 pandemic and the corresponding decision of the IPS President, all assessments will be carried out remotely – information and standards for distance assessment will be available on the MOODLE page of the UC.
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.
Assuming that the tests are carried out in person, the conditions of approval are:
1. if CF is greater than or equal to 10 and less than 17, the student is approved with a final grade equal to CF, provided that both test grades are greater than or equal to 7.0;
2. If CF is greater than or equal to 17, 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 16 values.
Test 1: December 17
Test 2: January 30
Exam Assessment
The exam assessment is in accordance with the usual rules (students who choose not to carry out the continuous assessment or fail do obtain approval on it may attend the regular exams), with the following additional conditions:
Le E be the grade obtained in the exam (rounded to the units),
1. if E is greater than or equal to 10 and less than 17, the student is approved with final grade E;
2. if E is greater than or equal to 17, 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 16 values.
Grade improvement
According to Article 11 of the IPS Student Performance Assessment Guidelines.
Comments:
The tests will have a duration of two hours and the exams of two and a half hours.
Bibliography
Bibliography
• Several notes and exercises made by teachers of the Mathematics Department (available on the Moodle page, in portuguese).
• James Stewart, Calculus, 4th edition, Brooks/Cole, 1999.
• E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 10th edition, 2011.
Complementary bibliography
• João P. Santos, Cálculo numa variável real, IST Press, 2016.
• Pedro M. Girão, Introdução à análise complexa, séries de Fourier e equações diferenciais, IST Press, 2014.
• Vasco Simões, Análise Matemática 2, Edições Orion, 2011.
• Azenha & M. A. Jerónimo, Elementos de cálculo diferencial e integral em IR e IRn, McGraw Hill, 2006.
• R. Larson, Calculus, 8th edition Vol. 2, McGraw-Hill, 2000.
• Gabriel E. Pires, Cálculo diferencial e integral em IRn, 2ª Edição, IST Press, 2014.
• T. M. Apostol, Calculus I and II, John Wiley & Sons, 1969.
Observations
• Enrolment for tests/exams is required up to one week before the date of the test/exam, on the Moodle page.
• It is mandatory to present an official identification document on tests and exams.
• The only consult forms allowed on tests and exams are the forms provided by the teachers (copies are available in the Moodle page).
• The handling or displaying of mobile phones or any other means of remote communication during the test is not allowed.
The office hours of the teachers will take place on the TEAMS platform and its schedule will be available on the Moodle page.
|
|
