Mathematical Analysis III

Code:

EM0015

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 Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LEM	0	Plano de estudos de transição para 2006/07	2	7	7	70	187
MIEM	248	Syllabus since 2006/2007	2	-	7	70	187
		Plano de estudos de transição para 2006/07	2	-	7	70	187

Teaching language

Portuguese

Objectives

1- BACKGROUND
The content and character of mathematics needed in engineering applications are changing rapidly. However Real analysis, ordinary and partial differential equations, remain indispensable. All the main fields of mechanical engineering are supported in mathematical analysis. For example Vibration Analysis subject needs the knowledge of Fourier Analysis and Partial Differential Equations. Another example is Solid Mechanics subject which needs the knowledge of Multivariable Integral Calculus.

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 develop 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%

Characterization of objectives and program
The objectives are to understand and dominate current techniques used for integration of differential equations (ordinary and partial differential equations) and systems of differential equations. This includes physical and geometrical interpretation and also integration over curves and surfaces in three dimensions. The concepts and applications of trigonometric polynomial approximations and Fourier series are also considered a main objective
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

5- LEARNING OUTCOMES
Knowledge and Understanding
On successful completion of the module, the student will be able to:
Understand the mathematics principles used in the field of mechanical engineering.
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.

Engineering Analysis –
On successful completion of the module, the student will be able to:
Obtain the solution of problems using mathematical analysis.
Apply knowledge and understanding to solve problems using the mathematical approach.
Select and apply relevant mathematical methods.

Program

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.

Mandatory literature

Apostol, Tom M.; Calculus. ISBN: 84-291-5001-3
luísa madureira; problemas de equações diferenciais e transformadas de Laplace, FEUPedições, 2000. ISBN: 972-752-040-5
Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-59989-1

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	70,00
	Teste	55,00
	Exame	5,00
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
	Estudo autónomo	59
	Total:	59,00

Calculation formula of final grade

exam

Examinations or Special Assignments

not considered