Subject: Mathematical Models

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

O1 - Approach (aspects of) mathematical models employed in various areas of natural sciences, humanities and engineering. Show how many of the concepts and techniques acquired throughout the degree in Mathematics arise in different areas and/or in different contexts of science and engineering.
O2 - Show that although there are problems that are seemingly different, they are somehow linked because they share a similar mathematical structure.
O3 - Consolidate some of the concepts acquired during the degree.

# Syllabus:

P1 - Mathematical fundamentals of Physics: as a general rule physical theories are based on laws (relations between quantities obtained from experience) that are assumed to be true. With the consolidation of the mathematical model of the theory it is often possible to show that what initially appeared to be a law is in fact a mathematical result obtained from other more basic laws (ie, more fundamental) of the theory. Mathematical concepts to be addressed: inner product and outer product, exact differential forms (conservative fields), gradient, divergence, curl and Laplacian. Theorems of Stokes, Ostrogradsky and Green, among others.
P2 - Applications of Linear Algebra involving matrices, eigenvalues and eigenvectors, determinants, linear applications, systems of linear equations and linear regression. Application of these concepts in the analysis of electrical circuits, traffic flow, problems within the social sciences, chemistry, geometrical optics, computer science, economics and biology.
P3 - Application of differential equations and special functions: show concrete examples of problems that involve solving differential equations (the logistic equation, Maxwell's equations, wave equation, Schrödinger equation, Bessel equation, Legendre equation and others) and manipulation of special functions (Heaviside function, Gamma function, Beta function, Bessel functions, Riemann zeta function, ...). Examples of practical applications in Biology, Demography, Economics, Chemistry, Sociology, Circuit analysis, Electromagnetism, Telecommunications, Acoustics, Heat conduction, Quantum mechanics, Radioactive dating, Signal processing, Fluid Mechanics, Astrophysics, String theory among others.
P4 - The Mathematical Model of the Universe: The pillars of physics: Quantum Mechanics and General Relativity. The Big Bang model (model predictions). The Standard Model of Particle Physics (how can we predict the existence of particles not yet observed considering symmetry, the Higgs boson and the LHC). Unification Theories (relation with group theory).

# Literature/Sources:

Herbert Goldstein , 1980 , Classical Mechanics , Addison-Wisley Publishing Company
Roald K. Wangsness , 1986 , Electromagnetic Fields , John Wiley & Sons
Erwin Kreyszig , 1988 , Advanced Engineering Mathematics , John Wiley & Sons
Jeffrey R. Chasnov , 2009 , Mathematical Biology , The Hong Kong University of Science and Technology
Ray D'Inverno , 1993 , Introducing Einsteins's Relativity , Oxford Univ. Press.
P. A. Tipler & R. A. Llewellyn , 2003 , Modern Physics , W. H. Freeman Company
C. Cohen-Tannoudji, B. Diu & F. Laloe , 1977 , Quantum Mechanics , John Wiley & Sons
A. Fuente , 2000 , Mathematical Methods and models for economists , Cambridge University Press

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Two tests (one at the middle and one at the end of the semester), each one with a weight of 35%, and a practical work (in report or article format), with a weight of 30%, to be delivered at the end of the semester. We opted for two tests taking into account that the syllabus is divided into four units. So the first test contemplates the two first units (first 35 hours of contact) and the second test contemplates the two last units (remaining 45 hours of contact). The work is to motivate students to do research and get results on their own initiative (students can always count on the support / guidance of the teacher). The work also has the purpose of creating good writting habits (which is not possible with an evaluation using tests only).