Mathematical Analysis II

Code:

L.EIC007

Acronym:

AM II

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Informatics and Computing Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EIC	449	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

1- BACKGROUND The main aim is to introduce fundamental mathematical concepts by developing the ability to analyze problems and results and also to acquire mathematical precision. These aspects form an educational background for other subjects in the curricula.

2- SPECIFIC AIMS Enhance the students reasoning capacity and knowledge of essential mathematical concepts. The students should acquire solid theoretical and practical training on the main concepts and results of differential and integral calculus of several variables, including the basic theorems of calculus.

3- PREVIOUS KNOWLEDGE Functions and graphs. Differential and integral calculus in R1. Vector algebra. Lines and planes in R3.

4- LEARNING OUTCOMES Knowledge and understanding: Partial and directional derivatives for real-valued and vector-valued functions; gradient vector. The chain rule for real-valued and vector-valued functions including implicit functions. Apply parametric curves and surfaces in R3 to calculate line and surface integrals. Establishment of the relationship between the line integral and the surface integral based on the Green’s, Stokes and Gauss Theorems.

 

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.  Students should be able to apply these concepts in engineering problems

Working method

Presencial

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

The knowledge related to differential and integral calculus acquired in the Mathematical Analysis I course, and the knowledge related to vector algebra and analytic geometry acquired in the Algebra course, both taught in the 1st Year, 1st Semester of the Computer Science and Engineering Bachelor's Degree program, are considered essential for the completion of this course.

Program

1- VECTOR FUNCTIONS. Properties. Curves. Arc length. Curvature.
2- FUNCTIONS OF SEVERAL VARIABLES. Quadric surfaces. Level curves and level surfaces. Partial derivatives. Limits and continuity. 
3-GRADIENTS. Differentiability and gradient. Gradients and directional derivatives. The mean-value theorem. Chain rules. Maximum and minimum values. Differentials.
4- DOUBLE AND TRIPLE INTEGRALS. The double integral over a region. Evaluating double integrals using polar coordinates. Triple integrals. Cylindrical coordinates. Spherical coordinates.  Jacobians; changing variables in multiple integration.
5- LINE INTEGRALS AND SURFACE INTEGRALS. Line integrals. Properties. Line integrals respect to arc length. Green’s theorem. Parametrized surfaces. Surface área. Surface integrals. Divergence and curl. The divergence theorem. Stokes’s theorem

Mandatory literature

SALAS-HILLE-ETGEN;CALCULUS-ONE AND SEVERAL VARIABLES-WILEY
José Augusto Trigo Barbosa; Noções sobre Geometria Analítica e Análise Matemática, Efeitos Gráficos, 2018. ISBN: 978-989-99559-7-4
José Augusto Trigo Barbosa; Noções sobre Análise Matemática, Efeitos Gráficos, 2020. ISBN: 978-989-54350-4-3
José Augusto Trigo Barbosa; Apontamentos de apoio às aulas teóricas, 2017

Complementary Bibliography

Madureira Maria Luísa Romariz Universidade do Porto. Faculdade de Engenharia; Problemas de integrais de linha e superfície e de séries de Fourier. ISBN: 978-989-99559-2-9
ERWIN KREYSZIG; ADVANCED ENGINEERIG MATHEMATICS-WILEY

Teaching methods and learning activities

Theoretical classes will be based on the presentation of the themes of the course unit. These classes are aimed to motivate students, where examples of application will be showed. Theoretical-practical classes will be based on the analysis and on problem solving by students, where they have to apply tools and mathematical concepts taught in theoretical classes. These classes are aimed to assess students’ understanding and dexterity of the themes of the course unit.

Attendance in practical courses is controlled and the student may not exceed the designated number of absences (25% of the scheduled hours) indicated by the professor for each practical course. If the student exceeds the specified number of absences, he/she will not be able to attend the course or take any exam in that course unless he/she has a special statute (see FEUP's pedagogical and evaluation rules).

keywords

Physical sciences > Mathematics > Mathematical analysis > Functional analysis

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	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

Participation in 75% of the recitation classes

Calculation formula of final grade

Classification improvement

Students who pass the tests may improve their scores on a final exam that covers the entire course.