Mathematical Analysis II

Code:

EC0006

Acronym:

AMAT2

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 2S

Active?	Yes
Web Page:	http://moodle.fe.up.pt
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	220	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 rational 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.

Learning outcomes and competences

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

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.

Working method

Presencial

Program

Programm
1. Differenciation
1.1. Geometry of scalar real-valued functions: the graph of a function; level curves and level surfaces; sections and their graphs.
1.2. Topology in R^n: Open set; disk, ball and neighborhood of a point, boundary point. Limits and continuity: Limit of a functio in na open set. Continuity theorems.
1.3. Differentiation: Partial derivatives. Linear approximation and tangent plane to the graphic of a function at a point. Derivative of a scalar function at a point. Differential, Jacobian and matrix of partial derivatives. Vector function. Derivative of a vector function at a point. Curl and divergence of a vector field and its physical interpretation.
1.4. Properties of the derivative.
1.5. Gradients and directional derivative.

2. Higher order derivatives; maxima and minima
2.1. Taylor’s theorem.
2.2. Extrema of real-valued functions: Critical Points, points of local maxima and local minima. Saddle point. First derivative test. Hessian matrix. Compact set and Absolute extrema..
2.3. Constrained extrema.
2.4. Implicit differentiation.

3. Double and triple Integrals
3.1. Regions of integrationin  R^2 and R^3.
3.2. Definition of na integral (Riemann); Fubini’s theorem.
3.3. Double integraI of a scalar function over a non-empty region of the plane.
3.4. Triple integraI of a scalar function over a non-empty region of the space.
3.5. Change of variables formula. Polar coordinates and cilindrical and sphererical coordinates.
3.6 Applications: Center of mass of a body and moments of inertia about the coordinate axis.

4. Curves and e Surfaces
4.1. Curves in the plane and in the space: parametrizations of a curve; velocity vector and tangente vector: tangent line to a curve; regular parametrization;
4.2. Path integral; length of a curve.
4.3. Line integral; Integration of vector fields and physical interpretation: work done by a force field;
4.4. Surfaces:  surfaces with no boundary and boundary of a surface, parametrized surfaces; tangent plane and normal vector to the surface at a point; regular parametrization; oriented surfaces;
4.5. The Integral of a scalar function over a surface. Area of a surface.
4.6. Surface integral and its physical interpretation: (flux across a surface per unit of time)

5. The integral theorems of vector analysis
5.1. Curl of a vector field in the plane and  Green’s theorem (applications).
5.2. Stokes theorem (applications).Physical interpretation: circulation of a vector field, irrotacional and conservative fields.
5.3. Gauss’s theorem (applications). Physical interpretation: incompressible fluids.

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:

This curricular unit introduces fundamental concepts related to the analysis of functions of several variables. It develops the ability to analyze problems and results and acquire mathematical precision. It induces an educational background for other subjects in the subsequent curricular units. The syllabus complements the learning obtained in the curricular unit of Mathematical Analysis 1.

Mandatory literature

Jerrold E. Marsden, Anthony J. Tromba; Vector calculus. ISBN: 0-7167-1856-1
Miroslav Lovric; Vector Calculus, Addison Wesley, 1997. ISBN: 0-201-42797-4

Complementary Bibliography

Jerrold E. Marsden, Anthony J. Tromba; Study Guide for Vector Calculus. ISBN: 0-7167-1980-0
Tom M. Apostol; Cálculo. ISBN: 84-291-5016-1 (vol. 2)
Acilina Azenha, Maria Amélia Jerónimo; Elementos de cálculo diferencial e integral em IR e IRn. ISBN: 972-8298-03-X

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.

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:

The focus is on coordination between the fundamental theoretical knowledge and the developments required in the following curricular units, being promoted the intuitive understanding of the concepts and calculation capabilities. It is intended to develop expertise in differential and integral calculus of real and vectorial functions of several variables, being able to apply knowledge and comprehension to solve problems in new situations, in broad multidisciplinar contexts, being able to integrate acquired knowledge.

Software

maple

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Teste	100,00
Total:	100,00

Eligibility for exams

Achieving final classification requires compliance with attendance at the course unit, according to the MIEC assessment rules. It is considered that students meet the attendance requirements if, having been regularly enrolled, the number of absences of 25% for each of the classes’ types is not exceeded.

Calculation formula of final grade

Evaluation consists on 2 written assessments (see 1st year calendar). All assessments are compulsory. Not being present in one assessment  implies a grade of "0" in the correspondent grading.

Result of the evaluation at the end of the semester:

CF= final mark
CT1= first written test' mark
CT2= second written test' mark

CF = 0,50*CT1+0,50*CT2

Students that were admited for evaluation at the end of the semester but did not succeed are admited to the exam of appeal.
.

Observations

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

Working time estimated out of classes: 4 hours