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Math Help - Vector spherical harmonics

  1. #1
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    Vector spherical harmonics

    I am looking for either a paper or a book to show that Vector Spherical Harmonics form a complete set. I only need it for Euclidean 3 space. As I have been searching I have noticed that they are used in a lot in physics so If anyone has seen them in a mathematical physics book that would be awesome. I can seem to find many math books that talk about Spherical harmonics very much.

    So far a I have used these references for Scalar Spherical Harmonics

    Analysis of spherical symmetries in Euclidean spaces by Claus Muller
    Pure and Applied mathematics vol IX orthogonal functions bu Sansone , G
    and I have reference the standard Fourier analysis text by Stien and Wiess.

    None of these really talk about Vector Spherical harmonics. However, I
    I did find two journal articles in the Library

    The Theory of Vector Spherical Harmonics , E.L Hill 1954 and
    Vector spherical harmonics and their applications to magnetostatics, Barrera et al. 1984.

    This 2nd paper claims that they are complete but does not supply the proof and a rundown of its references has not turned anything up.

    Thanks

    TES.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    I am looking for either a paper or a book to show that Vector Spherical Harmonics form a complete set. I only need it for Euclidean 3 space. As I have been searching I have noticed that they are used in a lot in physics so If anyone has seen them in a mathematical physics book that would be awesome. I can seem to find many math books that talk about Spherical harmonics very much.

    So far a I have used these references for Scalar Spherical Harmonics

    Analysis of spherical symmetries in Euclidean spaces by Claus Muller
    Pure and Applied mathematics vol IX orthogonal functions bu Sansone , G
    and I have reference the standard Fourier analysis text by Stien and Wiess.

    None of these really talk about Vector Spherical harmonics. However, I
    I did find two journal articles in the Library

    The Theory of Vector Spherical Harmonics , E.L Hill 1954 and
    Vector spherical harmonics and their applications to magnetostatics, Barrera et al. 1984.

    This 2nd paper claims that they are complete but does not supply the proof and a rundown of its references has not turned anything up.

    Thanks

    TES.
    The only reference in my texts is Hill. I found this, but I suspect you already know what's in it.

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    The only reference in my texts is Hill. I found this, but I suspect you already know what's in it.

    -Dan
    Thanks for the link but this is the problem that I keep running into is there is a lot of reference material on Scalar Spherical Harmonics Spherical harmonics - Wikipedia, the free encyclopedia, but not very much of Vector spherical harmonics Vector spherical harmonics - Wikipedia, the free encyclopedia

    I didn't even know that they existed and kept running into problems when I would try to expand vector fields component wise using the scalar harmonics. Trying to take the div, grad or curl of any of them was a nightmare. Using the VSH the vector calculus works great and operations like the divergence return expressions in terms of the scalar harmonics.

    Thanks again

    TES
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    A Plied Mathematician
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    Try Blatt and Weisskopf's Theoretical Nuclear Physics, Appendix B, and Morse and Feshbach's Methods of Theoretical Physics, Section 13.3. Those are references I found in Jackson's Classical Electrodynamics, 3rd. Ed., p. 449. Jackson uses vector spherical harmonics, but it doesn't look like he gives an exposition of their theory (proof of completeness, for example, which physicists are notoriously bad at providing - either that, or they just try to show they can get the Dirac Delta function from some linear combination of the functions).

    Cheers.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    Thanks for the link but this is the problem that I keep running into is there is a lot of reference material on Scalar Spherical Harmonics...
    Sorry about that. The search was for vector spherical harmonics. I guess Yahoo took some liberties with the search and I didn't check well enough.

    Quote Originally Posted by TheEmptySet View Post
    I didn't even know that they existed and kept running into problems when I would try to expand vector fields component wise using the scalar harmonics. Trying to take the div, grad or curl of any of them was a nightmare. Using the VSH the vector calculus works great and operations like the divergence return expressions in terms of the scalar harmonics.
    Perhaps this will help. I was thinking back and the first place I found out about them was in an Optics lab. Perhaps you can find a reference in Eugene Hecht's "Optics." I don't own the text and I suspect it probably doesn't cover the Mathematics end as comprehensively as you need, but then again it might. I had a long talk with him at one point and he at least seemed to be the type of man who dotted his i's and crossed his t's. It might be worth a look.

    Quote Originally Posted by Ackbeet View Post
    Try Blatt and Weisskopf's Theoretical Nuclear Physics, Appendix B, and Morse and Feshbach's Methods of Theoretical Physics, Section 13.3. Those are references I found in Jackson's Classical Electrodynamics, 3rd. Ed., p. 449. Jackson uses vector spherical harmonics, but it doesn't look like he gives an exposition of their theory (proof of completeness, for example, which physicists are notoriously bad at providing - either that, or they just try to show they can get the Dirac Delta function from some linear combination of the functions).

    Cheers.
    I guess Jackson's 3rd edition is more comprehensive than the 2nd on this. The 2nd edition doesn't do much with them.

    -Dan
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    Quote Originally Posted by topsquark View Post
    Sorry about that. The search was for vector spherical harmonics. I guess Yahoo took some liberties with the search and I didn't check well enough.


    Perhaps this will help. I was thinking back and the first place I found out about them was in an Optics lab. Perhaps you can find a reference in Eugene Hecht's "Optics." I don't own the text and I suspect it probably doesn't cover the Mathematics end as comprehensively as you need, but then again it might. I had a long talk with him at one point and he at least seemed to be the type of man who dotted his i's and crossed his t's. It might be worth a look.
    I have Hecht, 3rd Ed., and I'm pretty sure it does not go into them. There is a fourth edition out now, though.

    I guess Jackson's 3rd edition is more comprehensive than the 2nd on this. The 2nd edition doesn't do much with them.

    -Dan
    I think it is more thorough.
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