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Vector Laplacian

  1. Jul 19, 2010 #1


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    Tell me I'm not going mad. If I have a vector field of the form [tex]\mathbf{A}=(0,A(x,y,z),0)[/tex] and I want to take the Laplacian of it, [tex]\nabla^{2}\mathbf{A}[/tex], can I take the Laplacian of the co-ordinate function A(x,y,z)? Will this be the same for the case of cylindrical co-ordinates?

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  3. Jul 19, 2010 #2


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    That's true for Cartesian coordinates.
    For cylindrical coordinates, you must remember that a planar unit vector is dependent upon the angular variable, so take care to differentiate that unit vector as well.

    For a vector field [tex]\vec{A}=A(r,\theta,z)\vec{i}_{\theta}[/tex] we'll get, for example,
    where lower case on the scalar function indicates differentiation with respect to that variable.
    Note that this does NOT equal a naive application of the Laplacian operator merely to the scalar component A.
    Last edited: Jul 19, 2010
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