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Laplace and Poisson's equation.

  1. Dec 10, 2013 #1
    1. The problem statement, all variables and given/known data

    I want to verify for [itex]v=\ln( r)[/itex], that

    a)[itex]\nabla^2 v=0[/itex] for [itex]v[/itex] on a disk center at [itex](x_0,y_0)[/itex]. Therefore [itex]v[/itex] is Harmonic.

    b)[itex]\nabla^2 v=\frac{1}{r^2}[/itex] for [itex]v[/itex] on a sphere center at [itex](x_0,y_0,z_0)[/itex]. Therefore [itex]v[/itex] is not Harmonic.





    3. The attempt at a solution

    a) For circular disk, [itex]v=\frac{1}{2}\ln[(x-x_0)^2+(y-y_0)^2][/itex]
    [tex]\nabla v=\frac{\hat x (x-x_0)+\hat y(y-y_0)}{[(x-x_0)^2+(y-y_0)^2]^2}[/tex]
    [tex]\nabla^2 v=\nabla\cdot\nabla v=\frac{[(y-y_0)^2-(x-x_0)^2]+[(x-x_0)^2-(y-y_0)^2]}{[(x-x_0)^2+(y-y_0)^2]^2}[/tex]=0



    b)For a sphere, [itex]v=\frac{1}{2}\ln[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2][/itex]
    [tex]\nabla v=\frac{\hat x (x-x_0)+\hat y(y-y_0)+\hat z(z-z_0)^2}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2}[/tex]
    [tex]\nabla^2 v=\nabla\cdot\nabla v=\frac{[(y-y_0)^2+(z-z_0)^2-(x-x_0)^2]+[(x-x_0)^2+(z-z_0)^2-(y-y_0)^2]+[(x-x_0)^2+(y-y_0)^2-(z-z_0)^2]}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2]} [/tex]
    [tex]\Rightarrow\;\nabla^2 v=\frac{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2]}=\frac{1}{r^2}[/tex]


    So [itex]v[/itex] is Harmonic only for a two dimension disk, not in three dimension sphere.
     
  2. jcsd
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