(Created 2009-08-11.)
 MATHEMATICAL TOOLS FOR NANOSCIENCE EXTF20

Higher education credits: 7,5. Grading scale: TH. Level: G2 (First level). Language of instruction: The course will be given in Swedish. EXTF20 overlap following cours/es: FFF155, FFF155 och FFF155. Compulsory for: N2. Course coordinator: Peter Samuelsson, Fysiska inst (MN). Recommended prerequisits: FMA430 Calculus in Severable Variables, FAFA05 Physics - Waves, Thermodynamics and Atom Physics, Matlab. Assessment: Oral or written exam. Accepted lab reports. Accepted computer lab reports. Accepted excersize hand-ins. Home page: http://www.teorfys.lu.se/FFF155/.

Knowledge and understanding
For a passing grade the student must

* describe real and complex Fourier series and Fourier integrals,
define the Fourier coefficients and the different types of Fourier
spectra as well as in simple cases calculate the complex Fourier
coefficients and the Fourier integral.

* explain how to use the Fourier transform in optics, in image
processing and in studies of simple electrical circuits as well as
explain the concepts FFT, sampling, Nyquist frequency and aliasing.

* explain how to treat a system of first order, non-linear
differential equations as well as explain the concepts of fix-point,
linearization, eigenvalues and stability.

* describe what a transfer function is, how it can be realized with
operational amplifiers in simple cases as well as how an operational
amplifier can be used to obtain information on how the system reacts
on an harmonic in-signal.

* explain what a Bode-plot is and sketch a Bode plot in simple
examples as well as explain what feedback is and how it can be used
practically.

* formulate Gauss and Stokes theorems as well as derive, from Coulombs
law, the induction law and Amperes law, the corresponding Maxwellian
laws in integral and differential form.

* show that the, from Amperes law obtained Maxwellian law does not
conserve charge as well as how to complete Amperes law in order to
remedy this.

* show that Maxwell's equations in vacuum allows for solutions in the
form of electromagnetic waves traveling with the speed of light as
well as explain what properties the electromagnetic waves have.

* describe how and under what conditions one can modify Maxwell's
equations in order to take into account bound charges and currents in
a material medium.

* explain in a simple model how the refraction index and the
dielectric constant of a material are related as well as what it means
to an electromagnetic wave in a material that the refraction index has
an imaginary part.

Skills and abilities
For a passing grade the student must

* in simple cases use the Fourier integral and the Laplace transform
to solve differential equations.

* use Fourier technic for signal and image processing in simple cases
with the help of a computer.

* use a computer to numerically solve a system of coupled non-linear
differential equations.

Contents
From applications of physics and other parts of science, different mathematical and computational toals are introduced. Starting with specific problems, the methods are generalised and their universality is emphasized.

Mathematical tools that will be introduced are Fourier series and integrals, the Fourier transform, partial differential equations, equations of diffusion, linear systems, wave equation, Maxwells equation, vector analysis and Laplace transform.

Applications of these tools are introduced through a number of projects. These could involve e.g. electrical circuits, networks, filters, Harmonic signals, feedback systems, impedance, electromagnetism, diffusion, acoustics, musical instruments and mechanical systems.

Literature
The projects will be based on instructions produced at the department, L.Gislén.
Jönsson, P.: Matlab, Studentlitteratur.