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The objective of Matemática Aplicada is to learn key areas of mathematics that are necessary for understanding, modeling, predicting, identify and solve problems of general engineering and mechanical engineering in particular, providing powerful tools for a less elementary approach on the curricular units in engineering courses.

In each topic, students must acquire the following skills.

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 linear equations with constant coefficients with Laplace transforms.

4. Series

4.1. Numerical Series

Identify geometric, arithmetic and telescopic series, study their nature and calculate the sum when convergent.
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 Theorems.

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.

Knowledge acquired in the curricular units (UC) Mathematics I and Mathematics II of the current curricular plan. (Elements of Mathematics I and Elements of Mathematics II of the CTeSP are equivalent to Mathematics I) or in the UC Mathematical Analysis I, Analysis Mathematics II and Linear Algebra and Analytical Geometry from the previous curricular plan.

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 linear equations with constant coefficients with Laplace transforms.

4. Series

4.1.Numerical series

Convergent series, properties and convergence tests. Absolute convergence.

4.2. 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.

Matemática Aplicada (Applied Mathematics) 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.

The consolidation of knowledge by students will be based on reading the materials provided and autonomous resolution of exercises, 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.

Observation:

•  daytime classes will be on-site;


•  the class in after-work hours will be remotely, on the Microsoft Teams platform, in the Applied Mathematics team 2022/2023, in the specific channel of the class.

A 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. 

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 conditions of approval are the 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 is5greater 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

• In order 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 points), a student may recover one and only one of the tests, on the same day and time of the first date of exam.

• A student may recover a test even if he has not taken it or has dropped out.

• It will not be possible to recover a test with a view to improving the grade.

• To recover the test, the student will have 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.

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

Assessment rules (tests and exams)

• 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.


• The student can only leave the room one hour after the start and itimplies the final delivery of the test/exam.