Een coördinaatvrije basis voor vectorvelden in de context van rotatiesymmetrische operatoren met behulp van sferisch harmonische functies

Bachelor thesis (2021)

Authors

J. Fleuren Electrical Engineering, Mathematics and Computer Science

Contributors

R. van der Toorn Mathematical Physics - EEMCS (supervisor 1)
B. Janssens Analysis - EEMCS (supervisor 2)
Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright: © 2021 Jules Fleuren
Spherical harmonics Coordinate free Vector spherical harmonics Rotational symmetries
More Info
expand_more

Published Date

07-07-2021

Language

Dutch

Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

Vector spherical harmonics are a set of basis functions for vector fields derived from the spherical harmonic functions. They are commonly used in spectral methods in certain areas of applied mathematics. In most of the existing literature they are defined in a way that is heavily dependent on the used coördinate system. In this thesis we define vector spherical harmonics in a coördinate free manner. We first do this on a sphere, using two tensor operators: the exterior derivative and the Hodge-star. We then extend these functions to vector functions in 3-dimensional Euclidian space and define a last type of vector function to form a complete basis for vector fields. We do this in such a manner that the vector spherical harmonics inherit some of the nice properties of the spherical harmonic functions with respect to rotations and rotational symmetric operators.