Mathematical Analysis II

Code:

EM0010

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 2S

Active?	Yes
Web Page:	http://sigarra.up.pt/feup/pt/
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
MIEM	194	Syllabus since 2006/2007	1	-	6	65	162

Teaching language

Portuguese

Objectives

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. Computation of physical quantities.

Learning outcomes and competences

At the end of the semester students should be able to: 

1 - Using parametric representation of curves in Rn and get your vector tangent and normal; calculate line integrals along these curves.
2 - Discuss the continuity of scalar functions of several variables.
3 - Get directional and partial derivatives for scalar fields and vector fields and know how to build the gradient vector and Jacobian matrix.
4 - Calculate derivatives of composite functions of scalar fields and vector fields, as well as implicitly defined functions.
5 - Obtaining maximum and minimum of unconstrained functions of two or three variables and maximum and minimum of constrained functions, with one or two constraints, by the method of Lagrange multipliers.
6 - Calculate double integrals over limited regions of R2, either in Cartesian or polar coordinates and calculate areas of these regions. Relate a double integral to a line integral using Green's theorem.
7 - Calculate triple integrals over limited regions of R3, get the volume of these regions by integrating and transforming triple integrals in Cartesian integrals in cylindrical and spherical coordinates.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Knowledge acquired in courses of Mathematics and Mathematical Analysis I.

Program

I –Vector-valued functions depending on a real variable; parametric equations of a line in Rn. Limits, continuity, differentiation and integration of vector-valued functions and applications. Arc length evaluation. II - Introduction to surfaces in R3: quadric, cylindrical and revolution surfaces. General notions for real-valued functions of n variables: domain and graph. Vector-valued functions of n variables; parametrical representation for curves in Rn and surfaces in R3. Introductory topological notions on Rn. Limits and continuity for scalar and vector-valued functions of n variables. III - Differentiation: partial and directional derivatives; gradient vector; partial derivatives of higher order; total derivative or Fréchet’s derivative and differentiability of a scalar function of n variables. Applications of the gradient: tangent plane and maximum of a directional derivative. Differentiability of vector-valued functions of n variables–Jacobian matrix. Properties of the derivative; different cases of the chain rule. Functions defined implicitly; implicit function theorem and implicit differentiation. Taylor’s formula for scalar functions of n variables. Extrema of scalar functions of n variables; constrained extrema and Lagrange multipliers. IV - Double integrals: over a rectangle and over more general regions in R2. Properties and geometric interpretation of double integrals. Fubini’s theorem – changing the order of integration. Applications of double integrals to the computation of physical quantities. Changing variables in double integrals; double integrals in polar coordinates. Triple integrals: over rectangular parallelepiped and more general regions in R3 . Properties and geometric interpretation of triple integrals. Fubini’s theorem – changing the order of integration for triple integrals. Applications to volumes, average values, center of mass and moment of inertia. Changing variables: triple integrals in cylindrical and spherical coordinates. V – Line integral: definition, properties and applications. Green’s Theorem.

Mandatory literature

Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana , 2006. ISBN: 85-86804-56-8
C.C. António; Apontamentos de Análise Matemática II, 2017 (To be published by AEFEUP and on-line contents by SIGARRA)

Complementary Bibliography

Marsden, Jerrold E.; Vector Calculus, N. ISBN: 0-7167-1856-1
Apostol, Tom M.; Calculus, N. ISBN: 84-291-5001-3

Teaching methods and learning activities

Theoretical-practical classes are based on the presentation of the different themes of the course with the support of slides. All concepts and methodologies are exemplified by examples. Students are encouraged to participate in class by answering to questions asked by the professor individually or in group, as well as to solve exercises. Practical classes (2 hours per week) are based on problem solving. The exercises will be available on ‘Contents’. Professor will be available to answer to all questions and doubts.

keywords

Physical sciences > Mathematics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	97,00
Frequência das aulas	65,00
Total:	162,00

Eligibility for exams

Students cannot miss more classes than allowed by the Article 8 of General Evaluation Rules of FEUP.

Calculation formula of final grade

Final mark will be based on the following components: - 1st Test (T1)- It will cover the first half of the program of the course. Students will be informed about the date; if the student reaches a minimum mark of 8 out 20 it might account for 50% of the final mark according to the rules described below. - 2nd Test (T2) – Only for students who reached a minimum mark of 8 in the first test. It will cover the rest of the program of the course; if the student reaches a minimum mark of 8 out 20 it might account for 50% of the final mark according to the rules described below. - Final Exam (FE)- Simultaneous to the second test; It is for students who did not reach a minimum mark of 8 in the first test plus for students that although they reached the minimum mark of 8 in the first test decide to do the whole exam instead of the second test. This exam will cover all the program of the course. - Recurso Exam (RE)- For students who did not reach a minimum mark in both tests or who did not reach a passing grade in both tests or exam. It is also for students who want to improve their mark. It will cover all the program of the course. All the tests/exams will last 2 hours. Final Mark (FM) calculation: FM=( T1+ T2)/2 only if T1>=8 and T2>=8 . or FM= FE or FM= RE.

Examinations or Special Assignments

Not applicable.

Internship work/project

Não aplicável.

Special assessment (TE, DA, ...)

An exam, according to Articles 10 and 14 of General Evaluation Rules of FEUP.

Classification improvement

Students can improve their mark by attending to the recurso (resit) exam at the recurso season, according to paragraph 2 of Article 11 of General Evaluation Rules of FEUP.