Infinitesimal Analysis

Code:

M215

Acronym:

M215

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master's Degree in Network and Information Systems Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:CC	45	Plano de estudos de 2008 até 2013/14	2	-	7,5	-
			3
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-
MI:ERS	108	Plano de Estudos a partir de 2007	2	-	7,5	-

Teaching language

Portuguese

Objectives

Objectives:
Introduction to methods of solving ordinary differential equations with emphasis on equations and systems of linear differential equations.
Integral over path and Surfaces. Integral theorems of Vector Analysis.
Inverse function theorem and implicit function theorem and its main applications.

Learning outcomes and competences

Problem-solving skills. Theoretical understanding

Working method

Presencial

Program

Differential equations. 1st-order equations: Separable Equations, Exact equations, linear and Bernoulli equations Integrating factors. Linear equations. Existence and uniqueness theorems. Theory of solutions of linear equations.General solution of linear equation. Equations with constant coefficients. Solutions of the homogeneous equation. Methods for determining particular solutions of the general equation: method of undetermined coefficients and variation of parameters.
Ordinary and singular points of equations of non-constant coefficients. Resolution by power series in the neighbourhood of ordinary points.
Laplace transforms. Transforms of some functions. Properties. Inverse Laplace transform. Heaviside function and its transform. Solving differential equations with discontinuous 2 th member. The Delta- Dirac "function". Systems of differential equations.The convolution integral.

Vector fields. Line integrals of scalar fields on the length of arc. Line integrals in the general case. Line integrals of vector fields. Conservative field gradients and rotational fields. Simply connected domains. Test for independence of path.
Green theorem. Applications.

Parametrized surfaces in three-dimensional Euclidean space. Surface integrals of scalar functions. Surfaces. Area of a surface. Integral of a vector field on a surface. Divergence Theorem (Gauss) and Stokes' theorem.

Inverse Function theorem and Implicit Function theorem.

Mandatory literature

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

Complementary Bibliography

Bronson Richard; Moderna introdução às equações diferenciais
Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
Boyce William E.; Elementary differential equations and boundary value problems. ISBN: 0-471-31999-6
Marsden Jerrold E.; Vector calculus. ISBN: 7167-4992-0
Swokowski Earl W.; Calculo com geometria analitica. vol. ii. 2ª ed. trad. ISBN: 85-346-0310-3
Madureira Luísa; Problemas de equações diferenciais ordinárias e transformadas de Laplace. ISBN: 978-972-752-124-1
Young Eutiquio C.; Vector and tensor analysis. ISBN: 0-8247-6671-7

Teaching methods and learning activities

* Lectures:
Exposure of the material of the program 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

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Eligibility for exams

Two written tests, each with a mark of 10. 

Students are required to obtain a minimum of 3  in each test and total of at least 9.5 adding the two test marks. 

A minimum of 9.5 in the supplementary exam is required for approval. 

An additional test, either written or oral, may be asked of students aiming at marks over 16 out of 20. 

 

 

 

 

 

 

 

Calculation formula of final grade

he final mark is the sum of the marks of the two tests. 
For students taking the supplementary exam the final mark is the result of the exam. 

An additional test, either written or oral, may be asked of students aiming at marks over 16 out of 20. 

Examinations or Special Assignments

1st Test - 2 November 2012
2nd Test - 21 December 2012

Special assessment (TE, DA, ...)

Written exam. 

An additional test, either written or oral, may be asked of students aiming at marks over 16 out of 20. 

Classification improvement

The classification can be improved by improving the outcome of the suplementary exam, according to the rules of the University.

Observations

Bibliography:
-Calculus with Analytic Geometry (vol2) - Earl W. Swokowski-McGraw-Hill
- Differential Equations and Their Applications - M. Braun. Springer-Verlag
- Elementary Differential Equations and Boundary Value Problems - Boyce, W., DiPrima - RC, Rio de Janeiro: Books Scientific and Technical Publishing SA, 2002
- Modern Introduction to Differential Equations, Richard Bronson, Schaum, McGraw-Hill
- Issues of ordinary differential equations and Laplace transforms - Luísa Madureira (FEUP editions)
- Vector Calculus, Jerrold E. Marsden, Anthony Tromba - Freeman and Company
-Vector and Tensor Analysis - Eutyches C. Young - Marcel Dekker, Inc.
- Differential Equations - R. Bronson, Gabriel Costa - Schaum´s Outline Series
-Marsden, J. Calculus III, 2nd ed., Springer-Verlag, 1985
- Differential Equations - R. Bronson, Gabriel Costa - Schaum´s Outline Series