Mathematics II

Code:

M192

Acronym:

M192

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2009/2010 - 2S

Active?	Yes
Responsible unit:	Departamento de Matemática Pura
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	117	Plano de estudos de 2008 até 2013/14	1	-	7,5	-
MI:ERS	202	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

To understand and be able to use the objects and results given in the contents.

Program

I - LINEAR ALGEBRA

1. Real vector spaces; subspaces; linear combinations; subspace spanned by a set; linear dependence and independence; bases and dimension; coordinates in a basis; construction of a basis from a set of generators and from a set of linearly independent vectors.
2. Linear maps between real vector spaces; general form of linear maps in R, R^2, R^3; kernel and image of a linear map; isomorphisms; isomorphic spaces; linear maps defined by the images of the elements of a basis; matrix of a linear application with respect to a pair of bases; change of basis matrix; matrix of the sum of two linear maps and of the product of a linear map by a number; matrix of the composition of linear maps; relation between two matrices o the same linear map; matrix of the inverse of a bijective linear map; determinant of an endomorphism; injective endomorphisms and determinants.
3. Eigenvalues and eigenvectors of an endomorphism; characteristic polynomial; eigenspaces; bases of eigenvectors; diagonalizable endomorphisms.
4. Scalar product in R^n; geometric interpretation in R^2 e R^3; orthonormal bases; Gram-Scmidt process; orthogonal projection of a vector onto a subspace; orthogonal complement of a subspace; spectral theorem; vector product in R^3.

II - VECTOR CALCULUS

1. Parametrized curves in R^n; image of a curve; reparametrizations; continuity and derivatives; properties; velocity vector; regular curves; tangent line to the image of a curve at a point; length of the image of a curve; arc-length reparametrization; acceleration, tangent acceleration and normal acceleration; curvature; normal unit vector; binormal unit vector; Frenet frame; Frenet formulas; torsion.
2. Real functions of several variables; general notions on vector functions (sum, product, composition, graph, affine functions, projections, component functions, level hypersurfaces); basic notions of topology in R^n (open sets, closed sets, boundary points, accumulation points, compact sets); limit of a function when X tends to a point; continuity.
3. Derivatives of functions of several variables; directional derivatives of a function at a point; partial derivatives; c1 functions; derivative of a function at a point; jacobian matrix; relation with directional derivatives; properties of derivatives; derivative of the composition of two functions; gradient of scalar functions; local maxima and minima of scalar functions; critical points; saddle points; second order partial derivatives; Hessian matrix and its use in classifying critical points; maxima and minima conditioned by a level hypersurface; Lagrange multipliers.
4. Multiple integrals; area of sets in R^2 and volume of sets in R^3; evaluation of multiple integrals in cartesian coordinates; Fubini's theorem; change of coordinates in multiple integrals; evaluation of double integrals in polar cooordinates; evaluation of triple integrals in cylindrical and spherical coordinates.

Evaluation Type

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	70,00
	Total:	-	0,00