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Laplacian of a vector function

  1. Nov 21, 2014 #1
    Problem: The vector function A(r) is defined in spherical polar coordinates by A = (1/r) er

    Evaluate ∇2A in spherical polar coordinates

    Relevant equation: I'm assuming I have to use the equation 1671 on this website

    But I haven't got a clue as to how I would apply it since, for example, I don't know what Aθ is. Any hint on how to get started would help.
     
  2. jcsd
  3. Nov 21, 2014 #2

    ShayanJ

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    Gold Member

    You're vector field has only a r component, other components are zero!
    [itex]
    \vec A=A_r \hat e_r +A_\theta \hat e_\theta+A_\varphi \hat e_\varphi
    [/itex]
     
  4. Nov 21, 2014 #3
    So it would just be ∇2A = ∇2Ar - 2(Ar/r2)?
     
  5. Nov 21, 2014 #4

    ShayanJ

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    Gold Member

    No. The Laplacian of a vector field, is a vector field. What you wrote, is only the r component of that vector field.
     
  6. Nov 21, 2014 #5
    Right, so will it be ∇2 A = (∇2Ar - 2(A2/r2))er? Or am I missing the other components?
     
  7. Nov 21, 2014 #6

    ShayanJ

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    Gold Member

    That's correct.
     
  8. Nov 21, 2014 #7
    EDIT: Oops, I meant the answer I got is (-2/r3)er
     
    Last edited: Nov 21, 2014
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