Code: | EC0011 | Acronym: | AMAT3 |
Keywords | |
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Classification | Keyword |
OFICIAL | Mathematics |
Active? | Yes |
Web Page: | http://moodle.up.pt |
Responsible unit: | Mathematics Division |
Course/CS Responsible: | Master in Civil Engineering |
Acronym | No. of Students | Study Plan | Curricular Years | Credits UCN | Credits ECTS | Contact hours | Total Time |
---|---|---|---|---|---|---|---|
MIEC | 151 | Syllabus since 2006/2007 | 2 | - | 5,5 | 60 | 145 |
OBJECTIVES:
This course aims to acquaint students with concepts and analytical, numerical and qualitative techniques, which are essential to study the behaviour of engineering problems modulated by differential equations.
SKILLS: - Technical knowledge of underlying sciences (differential equations); To know how to deal with new problems and unfamiliar situations in diverse and multidisciplinary contexts; To be capable of dealing with complex situations, finding solutions or giving an opinion in situations where information is limited or incomplete; To develop competences that lead to a life long learning process in a self oriented and autonomous way; To be capable of communicating and presenting knowledge clearly and unambiguously
LEARNING OUTCOMES Students must be able to: Solve First Order Differential Equations; Analyse the behavior of the solutions of first order autonomous differential equations;Solve Linear Equations of Higher Order; Solve First Order Linear Systems of Differential Equations; Draw the Phase Portraits of Linear Systems; Analyse the Critical Point Behavior;
PREVIOUS KNOWLEDGE Knowledge of Algebra, Mathematical Analysis I and Mathematical Analysis II.
1. Differential Equations [10%]
1.1 Mathematical Models and Differential Equations
1.2 Solutions and particular solutions
1.3 Slope Fields and Solution Curves. Numerical solutions.
2. First Order Differential Equations [30%]
2.1 Existence and Uniqueness of Solutions
2.2 Separable Equations
2.3 Linear First-Order Equations
2.4 Substitution Methods and Exact Equations.
2.5 Introduction to Qualitative Solutions of First Order Differential Equations
2.6 Velocity models; Ortogonal Curves.
3. Linear Equations of Higher Order [30%]
3.1 General Solutions of Linear Equations
3.2 Homogeneous Linear Equations
3.3 Homogeneous Linear Equations with Constant Coefficients
3.4 Nonhomogeneous Linear Equations
3.5 Mechanical Vibrations; Forced Oscillations and Resonance
4. First Order Systems of Differential Equations [15%]
4.1 Linear Systems of Differential Equations and Applications
4.2 Matrices and Linear Systems
4.3 The Eigenvalue Method andLinear Systems
4.4 Qualitative analysis: Stability and Phase Plane; Phase plane Portrait.
5 Non-linear Systems. [10%]
5.1 Equilibrium points.
5.2 Linearization of Non-linear Systems around an Equilibrium point; Phase plane Portrait.
5.3 Hamiltonian and Gradient Systems.
6. Laplace Transform Methods [5%]
6.1 Definition and properties. Heaviside and Dirac Delta functions.
6.2 Solving Initial Value Problems.
PERCENT DISTRIBUTION
Scientific component:80%
Technological component:20%
DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:
This curricular unit introduces fundamental concepts related to the study of differential equations with application to various phenomena and engineering problems. The syllabus complements the learning obtained in the curricular units of Mathematical Analysis 1 and Mathematical Analysis 2.
This course is mostly instructive and it has a special focus on mathematical formulation and engineering problems. There is going to be a relation between the essential theoretical knowledge of this course and the other courses of this degree. An intuitive understanding of the concepts, as well as computer skills will be valued. Subjects will be presented in a clear and objective way and examples of physical and geometrical nature will be given. This curricular unit is inserted in the Moodle platform, in order to enhance the discussion among all participants. In this platform, all students have access to every issue provided by the teachers and may strengthen their concepts by solving self-evaluation tests. Students will be encouraged to use software (Matlab and Maxima) and calculating machines.
DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:
The focus is on coordination between the fundamental theoretical knowledge and the developments required in the following curricular units, being promoted the intuitive understanding of the concepts and calculation capabilities. It is intended to develop expertise in differential equations calculus, being able to apply knowledge and comprehension to solve problems in new situations, in broad multidisciplinary contexts, being able to integrate acquired knowledge.
Designation | Weight (%) |
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Teste | 90,00 |
Trabalho prático ou de projeto | 10,00 |
Total: | 100,00 |
Designation | Time (hours) |
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Estudo autónomo | 39,00 |
Frequência das aulas | 48,00 |
Total: | 87,00 |
Achieving final classification requires compliance with attendance at the course unit, according to the MIEC assessment rules. It is considered that students meet the attendance requirements if, having been regularly enrolled, the number of absences of 25% for each of the classes’ types is not exceeded.
Two mini-tests, MAD1 and MAD2, and 3 Moodle activities.
All evaluation are expressed in a scale 0-20.
Final Mark (CF) is defined by:
CF = 0.4*CAD1 +0.5*CAD2+0.1MDL
CAD1– result of 1st moment of evaluation (MAD1);
CAD2 – result of 2nd moment of evaluation (MAD2);
MDL-classification obtained in 3 Moodle activities evaluated in the following scale:
Students that were admited for evaluation in the Regular Season but did not succeed are admited to appeal Season.
The distributed evaluation obtained in previous courses, is not valid.
NOTE: All students are enrolled in the course classified according to this method.
Students who, having frequency, do not get approval on the course unit, have access to the examination of appeal to this effect, and may opt to be assessed only on the matters concerning one of the mini-tests (CAD1 or CAD2) or to the whole matter (Final Exam).
Final Exam
Students approved according to CF classification, may apply to the appeal exam, repeating MAD1 or MAD2 or all the matters (Final Exam) of the curricular unit.
Working time estimated out of classes: 3 hours