Vector Analysis

Code:

M219

Acronym:

M219

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2012/2013 - 1S

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

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	0	Plano de estudos de 2008 até 2013/14	3	-	7,5	-
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:M	75	Plano de estudos a partir de 2009	2	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-

Teaching language

Portuguese

Objectives

To introduce, in a concrete way, the main results of Classical Analysis of svereal variables as well as the ones of Vactorial Analysis, emphasizing techniques specific to this area as well as their applications.

Program

1) The Euclidean sapce; the usual inner product, norm and metric. Basic properties.

2) Metric spaces. Definition and basic properties of: interior point, closure point, boundary point; interior, closure and boundary of a set; open sets and closed setes; neighbourhood of a point; bounded sets; convergent sequences and Cauchy squences, uniqueness of the limit; complete metric spaces; continuity; compact sapces; connected and path-connected spaces.

3) Spaces of functions. Pontwise and uniform convergence. Contracting maps. The Banach fixed point theorem. Parametric version of this theorem. Picard's iterative method.

4) The inverse Function Theorem. The Implicit Function Theorem. Implicit differentiation. Constrained extrema and Lagrange multipliers. The tangent space at a regular point of a level surface viewed as the space of velocity of curves contained in the suface level and passing though the given point. Some corollaries of these theorems.

5) Paths. The path integral of a scalar funtion. Line integrals. Conservative vector fields and gradient vector fields, relations between these two concepts ans construction of a potencial of a given conservative vector field defined on an open and connected set. Green's Theorem. Computing areas using Green's Theorem.

6) Parametrized surfaces. Regularity conditions on parametrizations. Regular parametrized surfaces. Tangent space, normal space, tangent plan and normal line; independency of these notions on the parametrizations. Graph surfaces and surfaces of revolution. Area of a regular parametrized surface; integral of a scalar funtion over a regular parametrized surface. Orientation of a surface. Surface integrals of vector functions.

7) The rotational and the divergence operators. Stokes' Theorem. Gass' Theorem. Interpreting the meaning of the rotational of a vector field at a given point; Interpreting the meaning of the divergence of a vectore field at a given point.

Mandatory literature

Marsden, Tromba; Vector Calculus, W. H. Freeman and Company, 1988. ISBN: 0-7167-1856-1
Marsden, Hoffman; Elementary Classical Analysis, W. H. Freeman and Company, 1993. ISBN: 0-7167-2105-8
Elon Lages Lima; Espaços Métricos, Projecto Euclides, 2003. ISBN: 85-244-0158-3

Complementary Bibliography

Elon Lages Lima; Curso de Análise vol. 2, Projecto Euclides. ISBN: 85-244-0049-8
Serge Lang; Calculus of Several Variables, Springer, 1987. ISBN: 0-387-96405-3

Teaching methods and learning activities

Explanation of the several theoretic topics on the blackboard . Strong interconnection between the theoretical and practical classes at the level of exercises and examples and of theoretical results.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	65,00
	Total:	-	0,00

Observations

all situations of evaluation beyond the two provided examinations will have a component of oral examination