Real Analysis II

Code:

M1037

Acronym:

M1037

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 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	135	Official Study Plan	1	-	9	84	243

Teaching language

Portuguese

Objectives

The student should know:  the basic concepts concerning numerical series and function series; the fundamental results concerning the analysis of multivariate functions and understand the concepts of partial derivative, gradient vector, local maxima and minima, tangent plane to the graph of functions of two variables; the student should also know the methods of multiple integration and use them to determine areas, volumes, etc, of bounded plane or space regions, using change of variables if necessary.

Learning outcomes and competences

The student should know: the basic concepts concerning numerical series and function series; the fundamental results concerning the analysis of multivariate functions and understand the concepts of partial derivative, gradient vector, local maxima and minima, tangent plane to the graph of functions of two variables; the student should also know the methods of multiple integration and use them to determine areas, volumes, etc, of bounded plane or space regions, using change of variables if necessary.

Working method

Presencial

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

Real Analysis and basic concepts of Linear Algebra and geometry.

Program

1. Parametrized curves.
Numerical series and power series.: convergence criteria. Sequences and series of real functions of a real variable: cimple convergence and uniform convergence; convergence criteria; series of continuous functions. Abel's theorem on power series.

2. Differential calculus of vector-valued multivariate functions.
Graphs of real-valued functions of two variables, contour lines of functions of two variables and level surfaces of functions of three variables. Open and closed subsets of R^n. Accumulation point and isolated point. Limits and continuity of functions. Directional derivatives and partial derivatives. Derivative function at a point of a real-valued multivariate function. Gradient vector and derivability. Tangent plane to the graph of a function of two variables. Interpretation of the gradient vector. Normal line and tangent hiperplane at a point on the level surface of a function. Higher order derivatives. Derivative function at a point  of a vector-valued multivariate function. Jacobian matrix. Derivation of composition of functions. Derivation of the inverse of a function. Leibniz's rule. Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes.  Lagrange multipliers.

3. Multiple integrals.
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem.  Integration and change of coordinates.  Volume and surface areas of solids of revolution.

Mandatory literature

Spivak Michael; Calculus. ISBN: 0-914098-89-6
Marsden Jerrold E. 1942-2010; Elementary classical analysis. ISBN: 0-7167-2105-8

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

Software

GeoGebra

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	60,00
Teste	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	159,00
Frequência das aulas	84,00
Total:	243,00

Eligibility for exams

There are no restrictions.

Calculation formula of final grade

There will be a test in the middle of the semester, worth eight points. A second test will take place during the normal exam period, will be worth twelve points and will be related to the subjects that were not tested in the first test. The exam at the second exam season will be worth twenty points. Those who wish to do so can take only the part of the exam corresponding to one of the tests and keep the grade for the other test.

In no other case will the student be allowed to replace part of the exam by a test.

Any special exam can be either an oral or a written exam. No part of these exams can be replaced by the score obtained in a test.

Grades above 17 will only be awarded after making an extra test.

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.
No part of these exams can be replaced by the score obtained in a test.

 

Classification improvement

Exam. For these students it will not be allowed to replace any part of the exam by any test.

Observations

Any student can be asked to do an oral examination in case there are some dougts about the written examination.