Mathematical Analysis III

Code:

EM0015

Acronym:

AM III

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2009/2010 - 1S

Active?	Yes
Web Page:	http://www.fe.up.pt/smat
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEM	291	Syllabus since 2006/2007	2	-	7	70	187
		Plano de estudos de transição para 2006/07	2	-	7	70	187

Teaching language

Portuguese

Objectives

In this course students should master techniques used for integration of differential equations (ordinary and partial differential equations) and systems of differential equations. Students should also be able to solve mathematical problems (physical and geometrical) using differential equations and determining their solution, and integrate curves and surfaces in three dimensions spaces. Moreover, students should also know how to use Fourier series on different applications.
The students should be able to:
-solve differential equations and systems
-solve line and surface integrals
-represent periodic functions in Fourier series
-solve some partial differential equations

Program

Differential Equations – definition of general solution
First Order Linear Equation
Uniqueness theorem for a solution
Homogeneous equations and orthogonal trajectories
Exact differential equations and integration factor
Linear differential equation
Non-linear Ricatti equation
Second order equations- reduction of order: when the dependent variable do not appear in the second order equation
Linear equations of order n: differential linear equations of homogenous order n
Differential linear equations of homogenous order n of constant coefficient – vector space of solutions
Non-homogeneous equations – Wronski’s method


Systems of differential equations – basic concepts and examples
Systems of first order linear differential equations and its relation with linear differential equations of order n
Systems of linear differential equations of first order with constant homogeneous coefficients: general solution
Systems of non-homogeneous linear differential equations of first order: method of constant variation

The Laplace Transform- definition
The s-shifting theorem and t-shifting theorem
Inverse Laplace transform
The convolution theorem
The Laplace transform of discontinuous functions
Application to differential equations of constant coefficients

Line integral- properties and applications
Curves and parameterization
Irrotational fields
Potential function
Conservative fields
Surface integral- properties and applications
Operators- gradient, rotational, compound operators, Laplacian
Green’s theorem
Stokes’ theorem
Gauss’ theorem
Fourier analysis – Fourier series: periodic functions, odd and even functions, non periodic functions, expansions.
Convergence

Partial derivate equations – basic principals, solution,
Equations of first order
Equations of second order with constant coefficient
Methods of determining a solution
Wave equation
Heat equation

Mandatory literature

Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3
Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-59989-1
Madureira, Luísa; Problemas de equações diferenciais ordinárias de Laplace. ISBN: 972-752-065-0

Complementary Bibliography

Wylie, C. Ray; Advanced engineering mathematics. ISBN: 0-07-113543-X

Teaching methods and learning activities

Theoretical and practical lessons. In the theoretical classes detailed deduction of all the chapters of the program is presented where deduction and abstraction is fundamental.
In the theoretical-practical classes the students are presented with problems to solve by themselves after examples are given

Evaluation Type

Evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Subject Classes	Participação presencial	90,00
2 tests	Exame	5,00
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
continuous study	Estudo autónomo	94
	Total:	94,00

Eligibility for exams

Students have to attend to 75% of the classes, if they do not, they cannot be admitted to exams, unless they have a special status (see- General Evaluation Rules of FEUP).

Calculation formula of final grade

50% first test and 50% second test

Examinations or Special Assignments

Not applicable

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

According to General Evaluation Rules of FEUP