Mathematical Analysis III

Code:

EC0011

Acronym:

AMAT3

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 1S

Active?	Yes
Web Page:	http://moodle.up.pt
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	173	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.

Learning outcomes and competences

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;

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

PREVIOUS KNOWLEDGE Knowledge of Algebra, Mathematical Analysis I and Mathematical Analysis II.

Program

1. First Order Differential Equations [40%]
1.1 Mathematical Models and Differential Equations
1.2 Solutions and particular solutions
1.3 Slope Fields and Solution Curves. Numerical solutions.
1.4 Existence and Uniqueness of Solutions
1.5 Introduction to Qualitative Solutions of First Order Differential Equations 
1.6 Separable Equations
1.7 Linear First-Order Equations
1.8 Substitution Methods and Exact Equations
1.9. Velocity models; Ortogonal Curves.

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 Nonhomogeneous Linear Equations
2.5 Mechanical Vibrations; Forced Oscillations and Resonance

3. First Order 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 andLinear Systems
3.4 Qualitative analysis: Stability and Phase Plane; Phase plane Portrait.
3.5 Non-linear Systems. Linearization of Non-linear Systems around an Equilibrium point; Phase plane Portrait. Hamiltonian and Gradient Systems.

4. Laplace Transform Methods [15%]
4.1 Definition and properties. Inverse transform
4.2 Transformation of Initial Value Problems
4.3 Heaviside and Delta functions.


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 without final exam

Assessment Components

Designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	39,00
Frequência das aulas	48,00
Total:	87,00

Eligibility for exams

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.

Calculation formula of final grade

 Two mini-tests, MAD1 e MAD2. All evaluation are expressed in a scale 0-20.

Final Mark (CF) is defined by:

CF = 0.5*CAD1 +0.5*CAD2

CAD1– result of 1st moment of evaluation (MAD1);
CAD2 – result of 2nd moment of evaluation (MAD2);

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 1: The assessment tests, MAD1 e MAD2, associated with classifications  CAD1 and CAD2, are madatory.

NOTE 2: 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).

Special assessment (TE, DA, ...)

Final Exam

Classification improvement

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.

Observations

Working time estimated out of classes: 3 hours