Mathematical Analysis III

Code:

EIG0045

Acronym:

AM III

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2007/2008 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LGEI	0	Plano de estudos de transição para 2006/07	2	6	6	70	160
MIEIG	99	Plano de estudos de transiçao para 2006/07	2	-	6	70	160
		Syllabus since 2006/2007	2	-	6	70	160

Teaching language

Portuguese

Objectives

The objectives are: to transmit to the students various techniques of integration of diferential equations and systems of diferential equations, including physical and geometrical interpretations and also integration in curves and surfaces in three dimensions. The concepts and applications of trigonometric polinomial approximations and Fourier series are also considered a main objective

Program

Introduction to differential equations: general classification, definition of solution and of boundary value problems. Ordinary differential equations of first order: the existence and uniqueness theorem; separable equations; homogeneous equations; linear equations (homogeneous and non homogeneous). Some problems modeled by first order equations: problems in mechanics, population dynamics. Exact equations and integrating factors. Non linear equations reducible to linear ones: the Bernoulli equation. Ordinary higher order differential equations reducible to lower order equations. Linear equations of order greater than one: general theory of homogeneous and non homogeneous linear nth order equations. Existence and uniqueness theorem. General solution for homogeneous linear equations with constant coefficients. Linear non homogeneous equations: the variation of parameters method. Systems of first order linear equations: introduction and its relation with an nth order linear differential equation. Some examples. Basic theory of systems of first order linear equations. Homogeneous linear equations with constant coefficients. Real or complex single eigenvalues case and repeated eigenvalues case. Fundamental matrices. The method of variation of parameters for non homogeneous systems. The Laplace transform: definition and existence conditions. Laplace transform of some basic functions using the definition. Main properties of Laplace transform: first and second translation theorems and the transform of the derivative. Inverse Laplace transform. Solution of initial value problems and of differential equations with discontinuous forcing functions, using the Laplace transform. Impulse functions and Dirac δ-function. The convolution theorem.
Line integral of scalar and vector functions. Independence of path. Work done by a force. Green’s Theorem. Surface integral. Area of a surface, mass, center of gravity, centroid and moment of inertia. Flux integral. Theorems of Stoke´s and Gauss. FOURIER ANALYSIS Fourier Series. Euler formulas. Even and odd functions, half-range expansions. Approximation by trigonometric polynomials and minimum square error.
PARTIAL DIFFERENTIAL EQUATIONS Equations of first order. General solution of linear equations. Second order equations. Solution by the method of factorization for homogeneous partial differential equations with constant coefficients. The wave equation: D’Alembert solution and separation of variables. The heat equation.

Mandatory literature

Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-50729-6
luísa madureira; problemas de equações diferenciais ordinárias e transformadas de Laplace, FEUPedições, 2004. ISBN: 972-752-040-5

Complementary Bibliography

Wylie, C. Ray; Advanced engineering mathematics. ISBN: 0-07-113543-X
Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3

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	70,00
	Teste	40,00
	Exame	5,00
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
	Estudo autónomo	47
	Total:	47,00

Calculation formula of final grade

exam

Examinations or Special Assignments

not considered