Mathematical Analysis II

Code:

EC0006

Acronym:

AMAT2

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2009/2010 - 2S

Active?	Yes
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	374	Syllabus since 2006/2007	1	-	7	75	187

Teaching language

Portuguese

Objectives

SPECIFIC AIMS
To introduce fundamental concepts in view of the analyse of functions of several variables. To develop the ability to analyse problems and results and acquire mathematical precision. To induce an educational background for other subjects in the curricula.

Development of the reasoning capacity of the students and knowledge of essential mathematical concepts. Students should get solid theoretical and practical formation on the main concepts and results of differential and integral calculus of several variables, including the basic theorems of calculus.

PREVIOUS KNOWLEDGE
EC0001: Functions and graphs. Knowledge of differential and integral calculus of one real variable.
EC0002: Knowledge of Matrix Algebra.


PERCENTUAL DISTRIBUTION
Scientific component:75%
Technological component:25%

LEARNING OUTCOMES
Knowledge and Understanding: Partial and directional derivatives for real-valued and vector-valued functions; gradient vector and jacobian matrix. The chain rule for real-valued and vector-valued functions including implicit functions. Maxima and minima of unconstrained functions of two and three variables and for constrained functions as well as using Lagrange multipliers method applied to functions with one or two constraints. Evaluation of double integrals over general regions in R2 using rectangular and polar coordinates. Parametric curves in Rn and normal and tangent vectors; to calculate line integrals on that curves. Calculation of triple integrals over regions on R3 using rectangular, cylindrical and spherical coordinates. Establishment of the relationship between the line integral and the double integral based on the Green’s theorem. Application of Stokes and Gauss Theorem.

Engineering analysis- Differential and integral calculus of several variables.
Engineering design- Engineering problems of multiple variables.
Investigations- Practical formation on the main concepts and results of differential and integral calculus in Rn.
Engineering practice- Computation of physical quantities.
Transferable skills- Knowledge of differential and integral calculus of real and vector functions of several real variables.

Program

Chapter 1 –DERIVATION
The geometry of scalar functions
Partial derivative; directional derivative; matrix derivative
Second-order Taylor polynomial
Hessian matrix and classification of critical points
Implicit derivation
Chain rule

Chapter 2- INTEGRATION
Double integral and areas of regions with non empty interior in IR2
Triple integral and volume of regions with non empty interior in IR3
Change of variables theorem: polar, cylindrical and spherical coordinates
Application of integration

Chapter 3- CURVES AND SURFACES
Curves: parameterisation and sketch of plane curves; first derivative of a vector to find the equation of the tangent line to a curve; length of a curve and integration of scalar functions
Surfaces: level surfaces and parameterisations; tangent plan and normal vector; surface area and integration of scalar functions

Chapter 4- VECTOR FIELDS
-Introduction to vector fields; Special examples: velocity fields and force fields (Newton’s second law)
-Integration of vector fields: integration along curves and physical interpretation (fluid circulation and work of a force field and kinetic energy); surface integration and physical interpretation
-Gradient of a scalar function and its properties;
-Vector field rotational: definition of rotational, plan rotation and Green’s theorem, Stoke’s theorem. Physical interpretation: Irrotational fluids and conservative fields
- Divergence of a vector field: definition of divergence; Gauss’ theorem. Physical interpretation: incompressible fluids.

Mandatory literature

Marsden, Jerrold E.; Vector calculus. ISBN: 0-7167-0462-5
Abreu,A. H. S.; Funções de variável complexa, ISTpress, 2009. ISBN: 978-972-8469-77-1

Complementary Bibliography

Maria do Carmo; Lições de Análise Matemática 2
Rodrigues, José Alberto; Curso de análise matemática-cálculo em Rn, Principia, 2008. ISBN: 978-989-8131-16-4
Mathews, J. H., Howell, R. W.; Complex Analysis for Mathematics and Engineering, Jones and Bartlett, 2001. ISBN: 0-7637-1425-9
Tom M. Apostol; Calculus. ISBN: 84-291-5001-3
Edward M. Purcell; Electricity and magnetism

Teaching methods and learning activities

It is essentially a formative subject, coordinating fundamental theoretical knowledge with some approaches which are necessary in the subjects placed ahead in the course. At this level it is important to develop intuitive concepts as well as computer skills. The concepts are exposed in a clear and objective way, making frequent use of examples of physical or geometrical nature. The use of the software Maple will be asked, as a working tool, namely through the execution of a practical project.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	80,00
	Total:	-	0,00

Eligibility for exams

According to General Evaluation Rules of FEUP (Article 4)

Calculation formula of final grade

The mark of the final exam.

Special assessment (TE, DA, ...)

Written examination.
SPECIAL RULES FOR MOBILITY STUDENTS:
Proficiency in Portuguese and/or English;
Previous attendance of introductory graduate courses in the scientific field addressed in this module;
Evaluation by exam and/or coursework(s) defined in accordance with student profile.

Classification improvement

According to General Evaluation Rules of FEUP (Article 10)

Observations

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Working time estimated out of classes: 4 hours