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marcusl

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"Maxwell's Equations" are pretty open ended. There are bases that are useful for solving certain problems.

1. For scalar potential and Green function problems in magneto- and electro-statics, cylindrical and spherical harmonics come to mind as particularly useful. There are two different cylindrical harmonic bases as well, those with cosine/sines and ordinary Bessels, and those with cosh/sinh and modified Bessels. There are different versions of the spherical expansions, too. The alternate expansions sometimes have particular advantages, see

http://physics.princeton.edu/~mcdonald/examples/EM/cohl_apj_527_86_99.pdf" [Broken]

and

http://iopscience.iop.org/0022-3727/19/8/001?ejredirect=migration"

for two interesting applications.

2. Similar harmonics exist of course for spheroidal and all the other 8 or so coordinate systems that are separable under Laplace's equation.

3. The harmonic bases are useful for radiation problems as well (e.g., multipoles). For full EM wave problems that include polarization, the vector forms of the harmonic bases are used. Vector spherical harmonics are the best known of these.

4. Sines and cosines (your Fourier bases) are useful for problems in rectangular coordinates.

1. For scalar potential and Green function problems in magneto- and electro-statics, cylindrical and spherical harmonics come to mind as particularly useful. There are two different cylindrical harmonic bases as well, those with cosine/sines and ordinary Bessels, and those with cosh/sinh and modified Bessels. There are different versions of the spherical expansions, too. The alternate expansions sometimes have particular advantages, see

http://physics.princeton.edu/~mcdonald/examples/EM/cohl_apj_527_86_99.pdf" [Broken]

and

http://iopscience.iop.org/0022-3727/19/8/001?ejredirect=migration"

for two interesting applications.

2. Similar harmonics exist of course for spheroidal and all the other 8 or so coordinate systems that are separable under Laplace's equation.

3. The harmonic bases are useful for radiation problems as well (e.g., multipoles). For full EM wave problems that include polarization, the vector forms of the harmonic bases are used. Vector spherical harmonics are the best known of these.

4. Sines and cosines (your Fourier bases) are useful for problems in rectangular coordinates.

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Since SR is a particular analysis of Maxwell's Equations regarding time and length (minus the conjectural things Einstein added) spinors and quaterions must be the most reduced form of transformations of variables with the Maxwell Equations.

Quaterions function in that way apparently because they encode symmetry properties which vectors do not. At least that is my understanding.

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