(Created 2009-08-11.)

MATHEMATICAL TOOLS FOR NANOSCIENCE | EXTF20 |

*Knowledge and understanding*

For a passing grade the student must

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

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.