Analysis

Code:

M1019

Acronym:

M1019

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2014/2015 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=3250
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:CC	7	Plano de estudos a partir de 2014	2	-	6	56	162
MI:ERS	12	Plano Oficial desde ano letivo 2014	2	-	6	56	162

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

Adams Robert A.; Calculus. ISBN: 0-201-82823-5
Boyce William E.; Equações diferenciais elementares e problemas de valores de contorno. ISBN: 9788521613121

Complementary Bibliography

Braun M.; Differential equations and their applications. ISBN: 0-387-90114-0
Bronson Richard; Moderna introdução às equações diferenciais
Marsden Jerrold; Calculus iii. 2nd ed. ISBN: 0-387-90985-0
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

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.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Calculation formula of final grade

The regular evaluation will be based on one test (06/11) and a final examination, the classification distributed over 10 points ​​in the test and 10 in the final exam. The final grade is the sum of the grades of the tests and exam, with successive roudning to centesimals, decimals, and units.

All registered students are admitted to the tests and exam.

There will be an extra exam available to any student who does not obtain approval according to the above scheme.

Examinations or Special Assignments

Classification improvement

The general evaluation rules apply.