Mathematical Analysis III

Code:

EIG0045

Acronym:

AM III

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 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
MIEIG	114	Syllabus since 2006/2007	2	-	6	70	162

Teaching language

Portuguese

Objectives

1- BACKGROUND Mechanical Engineering evolution shows that advanced mathematics is of main importance in present skills and research areas. 2- SPECIFIC AIMS Development of the reasoning capacity of the students and knowledge of advanced mathematics for engineers. Students should get solid theoretical and practical skills on the main concepts and results of differential and integral calculus of several variables and be able to develope some technological applications. 3- PREVIOUS KNOWLEDGE Functions, graphs, three-dimensional integration, differential and integral calculus and linear algebra. 4- PERCENT DISTRIBUTION Scientific component 75% Technological component 25% 5- LEARNING OUTCOMES Knowledge and understanding of: ORDINARY DIFFERENTIAL EQUATIONS Linear equations of first order, Riccati equation, homogeneous equations and orthogonal trajectories. Equations of higher order. Reduction of order: some examples. Linear equations of order n , homogeneous solution and particular solution: method of variation of parameters. Systems of differential equations. The Laplace Transform. The s-shifting theorem and t-shifting theorem, the convolution theorem. The Laplace Transform of discontinuos functions. Application to differential equations 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. Surfaces orthogonal to a family of surfaces. 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.

Learning outcomes and competences

Students should deepen their knowledge on the concepts of line, surface, double and triple, integrals and their applications and improve the knowledge of vector and scalar fields. They must have the ability to solve differential equations. Students should be able to apply these concepts in engineering problems

Working method

Presencial

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
Madureira, Luísa; Problemas de equações diferenciais ordinárias de Laplace. ISBN: 972-752-065-0
Apostol, Tom M.; Calculus

Complementary Bibliography

Wylie, C. Ray; Advanced engineering mathematics. ISBN: 0-07-066643-1

Teaching methods and learning activities

Theoretical classes will be based on the oral presentation of the themes of the course unit, where deduction and abstraction are essential to understand the program. In practical classes students will have to solve problems based on texts or on an exercise book. There will be a presence sheet in every practical class. Students cannot miss more than 25% of the classes. Otherwise they will not be admitted to exams, unless they have a special status (See General Evaluation Rules of FEUP).

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	0,00
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	90,00
Frequência das aulas	70,00
Total:	160,00

Eligibility for exams

If students miss more classes than allowed by the rules, they will not be admitted to exams. Unless they have a special status (See General Evaluation Rules of FEUP).

Calculation formula of final grade

Test 1- 50% + Test 2- 50%

Examinations or Special Assignments

Not applicable

Internship work/project

Not applicable

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

At recurso exam (resit) students who fail to pass can repeat the first or the second test (the best mark will be taken into account). However, they can take a final exam, which will cover all themes of the course unit.

The successful students can improve their marks at recurso exam (resit), taking a final exam covering all themes of the course unit. The maximum mark (20 in 20) can only be achieved through an oral examination.