Engelska Studiehandboken kurser

Dept of Mathematics

MAM603 Mathematics C 30.0 ECTS credits

DENNA SIDA FINNS OCKSÅ PÅ SVENSKA

General information about studying at Luleå university

TIMEPERIOD:


LANGUAGE: Swedish

EXAMINER
T Gunnarsson Univ lekt


PREREQUISITES
MAM602

COURSE AIM
The course aim is to give deeper knowledge and skills in some central mathematical theories and structures and to give training in independent mathematical work, to give a foundation for further studies, both in the form of courses and independent studies, in mathematics and subjects where mathematics is used, to give deeper insight in those parts of mathematics which are needed for independent application of mathematical methodology, to give such deeper knowledge that will benefit the teaching of mathematics at the gymnasium.

CONTENTS
The course Mathematics C, 20 credits, consists of a combination of courses, which can be choosen from the courses below, with the exception that one can not chose both the course Algebra 2 and the course discrete mathematics. Each semester only some of the couses are taught. For the "kandidatexamen" an independent work for 10 credits is required. It is also possible to do an independent work for 20 credits. This is required for the "magisterexamen". Courses 1. Real analysis 1, 5 credits 2. Real analysis 2, 5 credits 3. Complex analysis, 5 credits 4. Partial differential equations 2, 5 credits 5. Algebra 2, 5 credits 6. Discrete mathematics, 5 credits 7. Applied mathematics, 10 credits After agreement from the examiner it is also possible to take other courses. Contents of the courses: 1. Real analysis 1, 5 credits The course treats foundational notions of the analysis of vectorvalued functions, such as limits, continuity, differentability, existence of inverse functions and implicitly defined functions and uniform convergence of function sequences. This is done starting with the properties of the natural numbers and the geometry and topology of the n-dimensional euclidean space. Convex sets and manifolds are treated as are more general topological and metric spaces. A short introduction to lebesgues measures and the lebesgues integral is also given in the course. Literature W.Fleming: Functions of several variables, Springer-Verlag. 2. Real analysis 2, 5 credits In the course a more thorough introduction to the theory of integration is given. The lebesgues measure and the lebesgues integral are introduced and applied. Literature M.Adams, V.Guillemin: Measure theory and probability , Wadsworth & Brooks. 3. Complex analysis, 5 credits The course deals with the complex plane and its essential geometric and topological properties, analytic and harmonic functions, Cauchy's theorem and the integral formula, residue calculus, conformal mappings and various transform methods such as the Fourier- and Laplacetransforms and the Z-transform. These notions are applied to models of planar flows and fields, to the solution of boundary value problems and to problems which have essential importance for applications in the physics. Literature S.D. Fischer: Complex Variables, Wadsworth & Brooks. 4. Partial differential equations 2, 5 credits The course deals mostly with three main types of second order partial differential equations: the Laplace«s equation, the wave equation and the heat equation. Initial and boundary value problems and their physical interpretation are discussed. Existence, uniqueness and stability of solutions are studied as well as different methods for finding the solutions. For example the following is studied: the Dirichlet problem for the Laplace equation, Green functions, the Cauchy problem for the wave equation, the energy problem, the spreading of waves, vibrating strings and membranes, the heatconduction and the diffusion problems and the maximum principle. Literature E.C. Zachmanoglou, D.W. Thoe: Introduction to Partial Differential Equations with Applictions. Dover Publ. Inc.. 5. Algebra 2, 5 credits The course treats the general theory of algebraic structures such as groups, rings and fields. Notions like coset, ideal and isomorphism are given special emphacis. The course treats field extensions and possible geometric constructions. In particular it is shown why the classical problems "trisection of the angle", "the doubling of the cube" and the "squaring of the circle" are geometricaly unsolvable. Computer exercises with the program Magma are included in the course. Literature J.B. Fraleigh: A first course in Abstract Algebra. Addison-Wesley. (5th or latest edition) 6. Discrete mathematics The course treats the general theory of algebraic structures such as groups, rings and fields. Notions like coset, ideal and isomorphism are given special emphacies. These notions are applied to Boolean algebras, blockmodels, codes with a finite number of states, algorithms for networks, geometric constructions etc. Literature N.Biggs: Discrete Mathematics, Clarendon Press. 7. C-paper, 10 credits Some topic is chosen for deeper studies and special examination work. Topics can, for example, be chosen from the following areas: Linear algebra Mechanical systems The calculus of variations Stochastic processes, ergodic theory and information theory The history of mathematics Mathematical didactics The choice of topic has to be done in concensus with the examiner Literature The literature is chosen in cooperation with the examiner Work with the C-paper may not commence until MAM602 or a corresponding course is completed. The C-paper has to be presented at a seminar.

TEACHING
The teaching consists of lectures, computer exercises and tutorials. The course can in its entirety and in part be taught in english.

EXAMINATION
The examination consists of written and oral examination.
COURSE GRADE SCALE: G, VG

ITEMS/CREDITS

                                                            	15.0ECTS
Written exam                                                	7.5ECTS
Written exam                                                	7.5ECTS

COURSE LITTERATURE
See the individual courses.

REMARKS
Recommended prerequisites MAM602. For the C-paper the following holds: MAM602 or a corresponding course must be completed before work with the C-paper is started. Under 1996/97 the following courses will be given: 3. Complex analysis, 5. Algebra 2 and supervising for the C-paper (or CD-paper for those who choose to do a 20 credit paper)
Last modified : 97-09-25

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