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I Vector Laplacian

  1. May 20, 2016 #1
    according to this page https://en.wikipedia.org/wiki/Vector_Laplacian value of Vector_Laplacian is vector, but according to this page https://en.wikipedia.org/wiki/D'Alembertian value of Vector_Laplacian is scalar

    8a806b56e1a8af77aca1897fcd8ebf9c.png
    Is on of these pages wrong or I misunderstand it?

    I am asking because I want to know what does Δ2 equal to in this 1ad8dd4174eba9373c9235546a3264b8.png and this 5020dc7c1608d709fcad4d7db1f19b50.png equation on this https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Coulomb_gauge page.
     
  2. jcsd
  3. May 20, 2016 #2

    Orodruin

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    The d'Alembertian is a differential operator. You can apply it to a scalar and get a scalar, or you can apply it to a vector and get a vector.
     
  4. May 20, 2016 #3
    So since magnetic potential A in this equation 1ad8dd4174eba9373c9235546a3264b8.png is vector ##\nabla^2=(\sum_{i=1}^3(\frac{\partial^2 A}{\partial x_i*\partial x_1});\sum_{i=1}^3(\frac{\partial^2 A}{\partial x_i*\partial x_2});\sum_{i=1}^3(\frac{\partial^2 A}{\partial x_i*\partial x_3}))## not ##\nabla^2=\sum_{i=1}^3(\frac{\partial^2 A}{\partial x_i^2})##?
    These are Cartesian coordinates.
     
  5. May 20, 2016 #4

    Orodruin

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    It is unclear what you intend to write with these expressions. In particular, it is unclear what you intend for ##A## to be (the vector, the vector components, etc). I strongly suggest that you do not write vectors on component form (x;y;z), but instead use basis vectors. This will make it much clearer what is intended.

    The Laplace operator applied to a vector in Cartesian coordinates is such that the ##x## component of ##\nabla^2 \vec A## is equal to ##\nabla^2 A_x##, where ##A_x## is the ##x## component of ##\vec A##.
     
  6. May 20, 2016 #5
    Last edited: May 20, 2016
  7. May 20, 2016 #6

    Orodruin

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    Neither, see my previous post.

    Your first option is just one term in the sum, your second is grad(div(A)) and not the Laplacian of A, your third is a scalar.
     
  8. May 20, 2016 #7
  9. May 20, 2016 #8

    Orodruin

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  10. May 20, 2016 #9
    Ok, thanks.
     
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