Analysis III

Code:

M2009

Acronym:

M2009

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	2	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	60	Official Study Plan	2	-	6	56	162
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162
MI:EF	102	study plan from 2017/18	2	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Introduction to methods of solving ordinary differential equations with emphasis on equations and systems of linear differential equations. 
Study of surfaces, line integrals and surface integrals, and study of classical theorems of Vector Analysis: Green's theorem, divergence Gauss theorem and Stokes theorem.

Learning outcomes and competences

Problem-solving skills. Theoretical understanding

Working method

Presencial

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

Calculus in one and several real variables and Linear Algebra.

Program

A. Ordinary Differential equations (ODE). Study of the initial value prolems (IVP) for some types of ODE.

1. Reference to Cauchy-Lipschitz theorem on the existence and uniqueness of solutions of the IVP for systems of first order ODEs. Transformation of an arbitrary order system  to a first order system. 

2. Expliict Solutions of some  ODE: scalar 1st order linear equations.  separable equations, 1st order homogeneous equations, Bernoulli equations, Ricatti equations, exact differental equations.

3. Linear ODE. Existence and uniqueness theorems. Solutions of  the associated  homogeneous equations. Fundamental systems of solutions. Order reduction method. In case of inear ODE with constant coefficients use of the zeros of the characteristic polynomial to compute a fundamental system of solutions. Methods for determining particular solutions of the general equation: method of undetermined coefficients and variation of parameters. Exponential of a linear operator. Systems of linear ODE.

B.Vector Analysis

1. Paths in open subsets of Rn. Line integrals. Vector fields. Gradient of a scalar function, gradient and conservative vector fields. Conditions for a vector field to be a gradient vector field. Green's theorem.

2. Regular submanifolds of Rn. Regular parametrizations. Tangent space and Normal space at each point.

Orientation of compac regular surface. Opens subsets with regular boundary. Orientation of the boundary.

3. Surface integrals of scalar functions. Surface area. Divergence of a vector field. Flux of a vector field along a surface. Gauss theorem of divergence.  Suficient conditions for a scalar map to be a divergence.  Suficient conditions for a vector field to be a rotational.  Stokes theorem.

Mandatory literature

Carlos Menezes; Apontamentos de Análise-III-2020-2021 (The Lecture Notes will be sent to Moodle along the semester)

Complementary Bibliography

Braun M.; Differential equations and their applications. ISBN: 0-387-90114-0
Luisa Madureira; Problemas de equações diferenciais ordinárias e transformadas de Laplace
Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
Marsden Jerrold E.; Vector calculus. ISBN: 0-7167-0462-5

Comments from the literature

Lectures given in classes and the Lecture Notes  are the most important "bibliography"

Teaching methods and learning activities

Lectures: Detailed exposure of the program content  and resolution of exercises.
Pratical Classes: Resolution, by the students, of the proposed exercises and answering questions about the resolution of problems and proposed work.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	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

The attending of theoretic or practice class is not mandatory.

Calculation formula of final grade

The final classification will be the score obtained in the final exam.

Examinations or Special Assignments

Special assessment (TE, DA, ...)

 

Any type or special examination can be from one the following types: exclusively by an oral examination,  only  a written exam, one oral examination and a written exam.

The decision of which of the above types is each special examination  is exclusively the responsability of the teacher  assigned to the curricular unit.

Classification improvement