On the mathematical structure
of electromagnetic theory
Lauri Kettunen, Jari Kangas,
Timo Tarhasaari
Institute of Electromagnetics
One theory, different ”languages”:
Classical vector analysis
Differential forms
… or in terms of chains and cochains
Now, let’s ask:
Are these various forms of the theory ”equal”?
Different languages do represent the very
same theory, but they start from different
assumptions.
Vector analysis has its pedagogical advantages
in representing electromagnetism,
… but it starts from the metric structure
(The metric structure is what makes the approach
pedagogical; For, lengths, areas, volumes, norms, etc are all
in use.)
Modern approach
• …based on differential forms, (co)chains, etc.
is more abstract, for less structure is assumed
at the first place.
In computing …
• … the advantage is: less structure
results in more simple code
Basic idea:
Employ only the structure which is necessary.
As a result the underlying code is more generic and less
complicated to develop and maintain.
Recognition of the structure ...
• enables one to formalize and properly solve
certain questions (S. Suuriniemi’s talk)
• enables one to circumvent some technical
problems which are considered difficult (P.
Raumonen’s talk)
Recognition of the employed structure yields a better
understanding of the underlying physics behind
electromagnetism
The expected advantages has created a trend
of geometrization in electromagnetism
List of needs in computation:
1. A discrete counterpart to a smooth manifold M
2. A discrete counterpart to differential forms, i.e.
cochains
3. A discrete counterpart to integration, i.e.
evaluation of cochains on chains
4. A discrete coboundary d with discrete version
of Stokes theorem <du,c> = <u, bd c>
5. A discrete version of the global inner product of
forms [ , ]
6. A discrete Hodge operator
7. A discrete wedge product and volume form
8. A discrete adjoint coboundary map
9. All of this should converge to the continuum
theory in an appropriate
limit
10. These are all compatible with the expected
relations holding such as
[u,v] = int_M u/\*v= int_M (u,v) vol,
Ref: Discussion with J. Harrison, Univ. Berkeley and
A. Bossavit, Univ. Paris
Chainlet theory ...
• basically includes all this, but it is not yet
known, how chainlets should be capitalized in
finite element kind of computing
Chainlet theory, ref: J. Harrison