Real Analysis II

Code:

M1037

Acronym:

M1037

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	105	Official Study Plan	1	-	9	72	243
L:MA	109	Official Study Plan	1	-	9	72	243

Teaching language

Portuguese

Objectives

• Introduce the fundamental concepts and results of numerical series and of sequences and series of real functions with a real variable.


• Introduce the fundamental concepts and results of vector analysis.

Learning outcomes and competences

Upon completing this course unit, the student should know:

• the basic results and techniques on series of real numbers and sequences and series of real functions of a real variable, namely power series;


• the fundamental results of differential analysis of vector functions of vector variable, namely the notions of continuity, derivative at a point, partial derivatives, directional, gradient and tangent space to the graph, as well as the basic results and techniques related to the determination of extremes of scalar functions, in a open domain or conditioned;


• the concepts, basic results and calculation methods concerning multiple integrals.

Working method

Presencial

Program

1. SERIES

• Real number series and convergence criteria.


• Sequences and series of real functions of a real variable: simple and uniform convergence; convergence criteria; series of functions; continuity, differentiability and integrability of functions defined by series.

2. VECTOR ANALYSIS

• Basic concepts of R^n topology: open, closed, bounded, compact sets, accumulation points and isolated points.


• General concepts about vector functions of vector variable: graphs, level sets, limits and continuity.


• Differentiability: directional and partial derivatives; Jacobian matrix and derivative at a point; derived from the compound; gradient of scalar functions; space tangent to the graph of a function. 


• Maximums and minimums of scalar functions: higher order derivatives and Hessean matrix; local and global extremes; determination of the extremes of differentiable functions defined in open; conditioned extremums and Lagrange multipliers method.

3. MULTIPLE INTEGRALS

• Definition of the Riemann integral of a function (of several variables) over a rectangle and over a bounded region. Integrability conditions.


• Fubini's theorem and calculation of double and triple integrals.


• Change of coordinates theorem for multiple integrals. Polar, cylindrical and spherical coordinates.

Mandatory literature

Michael Spivak; Calculus. ISBN: 0-914098-89-6
Jerrold E. Marsden; Vector calculus. ISBN: 0-7167-0462-5

Complementary Bibliography

Jerrold E. Marsden; Elementary classical analysis. ISBN: 0-7167-0452-8

Teaching methods and learning activities

Contact hours are divided into theoretical and theoretical-practical classes. In the first, the contents of the program are presented, using examples to illustrate the treated concepts and guide the students. In theoretical-practical classes, previously indicated exercises and problems are solved. Support materials are available on the course page. In addition to classes, there are weekly service periods where students have the opportunity to clarify doubts.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation without 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	171,00
Frequência das aulas	72,00
Total:	243,00

Eligibility for exams

Attendance is not required.

Calculation formula of final grade

Special assessment (TE, DA, ...)

The exams required under special statutes will consist of a written test that may be preceded by an eliminatory oral test, to assess whether the student is in the minimum conditions to try to pass the subject in the written test.