Analysis III

Code:

M2009

Acronym:

M2009

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 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	1	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	75	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	64	study plan from 2017/18	2	-	6	56	162

Teaching language

Portuguese

Objectives

Introduction to methods of solving ordinary differential equations with emphasis on equations and systems of linear differential equations.  Regular surfaces of R^3, Line Integrals and Surface integrals. Classical theorems of Vector Analysis: Green's theorem, divergencetheorem and Stokes theorems.

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 C^1 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 with continupus coefficients. Existence and uniqueness theorems. Vector spaces of 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 domains of R^n. Line integrals. Vector fields. Gradeint of a scalar function, gradient and conservative vector fields, Convex, star-shaped and simply connected domains. Conditions for a vector field to be a gradient vector field.  Green's theorem.

2.  Regular submanifolds of R^n: inverse image of a regular value of a scalar function , regualr parametrizations, tangent space and normal space at a point  Orientation of compact regular  surface.  Open subsets with piecewiise regular boundary. Orientation of the boundary

3. Surface integrals of scalar functions, Surface area.  Flux of a vector field along a surface. Divergence of vector field.  Laplacian of a scalar function. Gauss theorem of divergence. Condtions for a scalar function to be a divergence. Rotational of a vector field in an oriented open subset of oriented  R^3. Comditons for a vectior field to be a rotational.  Stokes theorem.

Mandatory literature

Carlos Menezes; Apontamentos de Análise III-2018-2019

Complementary Bibliography

Braun M.; Differential equations and their applications. ISBN: 0-387-90114-0
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 is 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