Analysis III

Code:

M2037

Acronym:

M2037

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	48	162
L:CC	0	study plan from 2021/22	2	-	6	48	162
			3
L:F	51	Official Study Plan	2	-	6	48	162
L:G	0	study plan from 2017/18	2	-	6	48	162
			3
L:Q	0	study plan from 2016/17	3	-	6	48	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, Gauss and Stokes theorems.

Learning outcomes and competences

Problem-solving skills. Theoretical understanding,and application of the results and techniques introduced to Physics problems.

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 the 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. Gradient of a scalar function, gradient and conservative vector fields, Convex, star-shapeped sets. Conditions for a vector field to be a gradient vector field.  Green's theorem.

2.  Regular submanifolds of R^3: inverse image of a regular value of a scalar function, regular parametrizations, tangent and normal spaces at a point  Orientation of a regular  surface. Orientation of the boundary

3. Surface integrals of scalar functions, Surface area.  Flux of a vector field along a surface. Stokes and Gauss theorems.

Mandatory literature

Luísa Madureira; Problemas de Equações Diferenciais Ordinárias e Transformadas de Laplace, Quântica Editora, 2020. ISBN: 9789898927583
Martin Braun; Differential equations and their applications. ISBN: 0-387-97894-1
Jerrold E. Marsden; Vector calculus. ISBN: 978-1-4292-2404-8

Comments from the literature

The lecture notes are also part of the "main bibliography".

Teaching methods and learning activities

Theoretical classes: Exposition of the program subjects. Exercise proposals.

Practical classes: Resolution of part of the proposed exercises and clarification 
of doubts about solving problems and proposed work.

Autonomous theoretical and practical study is strongly encouraged.

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	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

The attending of theoretic or practice class is not mandatory.

Calculation formula of final grade

The final grade will be the grade obtained from the final exam rounded to the nearest integer.

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 option for one of the alternatives is exclusively up to the jury of the curricular unit.

 

Classification improvement