Mathematical Analysis III
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2011/2012 - 1S
Cycles of Study/Courses
Acronym |
No. of Students |
Study Plan |
Curricular Years |
Credits UCN |
Credits ECTS |
Contact hours |
Total Time |
MIEC |
345 |
Syllabus since 2006/2007 |
2 |
- |
5,5 |
60 |
145 |
Teaching language
Portuguese
Objectives
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:
Skills to be developed according to CDIO (Conceive- Design- Implement- Operate- an engineering education initiative that was formally founded by Massachusetts Institute of Technology) :
- 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
1- solve First Order Differential Equations
2- draw the slope field and solution curves
3- solve Linear Equations of Higher Order;
4- solve First Order Linear Systems of Differential Equations;
5- draw the Phase Portraits of Linear Systems;
6- analyse the Critical Point Behavior;
Program
1. First Order Differential Equations [40%]
1.1 Mathematical Models and Differential Equations
1.2 Solutions and particular solutions
1.4 Existence and Uniqueness of Solutions
1.3 Introduction to Qualitative Solutions of First Order Differential Equations
1.5 Slope Fields and Solution Curves
1.6 Separable Equations
1.7 Linear First-Order Equations
1.8 Substitution Methods and Exact Equations
1.9. Mathematical Models and Numerical Methods: Euler’s Method and the Runge-Kutta
2. Linear Equations of Higher Order [30%]
2.1 General Solutions of Linear Equations
2.2 Homogeneous Linear Equations
2.3 Homogeneous Linear Equations with Constant Coefficients
2.4 Mechanical Vibrations
2.5 Nonhomogeneous Linear Equations
2.6 Forced Oscillations and Resonance
3. First Order Linear Systems of Differential Equations [15%]
3.1 Linear Systems of Differential Equations and Applications
3.2 Matrices and Linear Systems
3.3 The Eigenvalue Method and
4. Introduction to Qualitative Solutions of First Order Differential Equations [15%]
4.1 Stability and Phase Plane
4.2 Linear and Almost Linear Systems
4.3 Linearization Near a Critical Point
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.
Mandatory literature
Maria do Carmo Coimbra; Equações diferenciais: uma primeira abordagem, 2009
C. Henry Edwards, David E. Penney;
Differential Equations and Boundary Value Problems. Computing and modeling
Complementary Bibliography
Colecção de exercícios, AM3, MIEC, 2009
George F. Simmons, Steven G. Krantz ; trad. Helena Maria de Ávila Castro;
Equações diferenciais. ISBN: 978-85-86804-64-9
Figueiredo, Djairo; Neves, Aloisio; Equações Diferenciais Aplicadas, IMPA, 2002. ISBN: 85-7028-014-9
Stewart, James 1908-1997;
Cálculo. ISBN: 85-211-0484-0
Teaching methods and learning activities
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.
Software
Maxima
keywords
Physical sciences > Mathematics > Mathematical analysis > Differential equations
Evaluation Type
Distributed evaluation with final exam
Assessment Components
Description |
Type |
Time (hours) |
Weight (%) |
End date |
Attendance (estimated) |
Participação presencial |
52,00 |
|
|
Pratical Examination |
Teste |
4,00 |
|
2011-12-16 |
Examinations |
Exame |
6,00 |
|
2012-02-10 |
|
Total: |
- |
0,00 |
|
Amount of time allocated to each course unit
Description |
Type |
Time (hours) |
End date |
Study time to follow lessons |
Estudo autónomo |
36 |
2011-12-16 |
Study time with professors |
Estudo autónomo |
20 |
2012-02-10 |
Study time for examinations |
Estudo autónomo |
27 |
2012-02-10 |
|
Total: |
83,00 |
|
Eligibility for exams
According to General Evaluation Rules of FEUP, students must attend to 75% of the classes
Calculation formula of final grade
E: grade of the final exam
A: grade of the practical assignment
Final Mark = max { E, 0.8 E + 0.2 A}
Special assessment (TE, DA, ...)
Final Exam
SPECIAL RULES FOR MOBILITY STUDENTS:
Proficiency in Portuguese; Previous attendance of introductory graduate courses in the scientific field addressed in this module; Evaluation by exam and/or coursework(s) defined in accordance with student profile.
Classification improvement
Final Exam
Observations
PREVIOUS KNOWLEDGE
Knowledge of Algebra, Mathematical Analysis I and Mathematical Analysis II.
............................................................
Working time estimated out of classes: 3 hours