General overview:
The aspects of potential theory presented in this lecture mainly deal with how fields can be described by potentials, what properties of the fields follow from these descriptions, and how we can use these properties to calculate or model fields when only a limited amount of information is available by measurements. The examples are mostly centered around magnetic fields in space; however, the same theory can also be used for electric and gravity fields, and it can very similarly be applied to problems in, e.g., applied geophysics, geology, or astrophysics.
List of topics (topics at the end of list if time permits):
- Introduction
- Laplace and Poisson equations, harmonic functions, Green's
identities, Helmholtz theorem
- Green's functions as solutions of Dirichlet's and Neumann's boundary
value problems
- Field continuation, component transformation, and separation of
internal and external parts:
a) Cartesian geometry (methods based on Green's functions, Hilbert
transformation, Fourier space methods)
- Spherical and spherical cap harmonics
- Field continuation, component transformation, and separation of
internal and external parts:
b) Spherical geometry (methods based on Green's functions, spherical
harmonic methods)
- Concept of equivalent currents in cartesian and spherical
coordinates, current function, description of currents in space
using spherical harmonics
(outside an inside of source regions)
- Spherical elementary systems and their application to field
continuation
- Toroidal and poloidal fields and their properties
- Application examples for deriving ionospheric/magnetospheric
electrodynamics using methods that involve potential theory