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Understanding the manipulation of Laplacian

  1. Jun 25, 2009 #1
    Hi;

    I am trying to understand the rytov approximation... and when I was studying that, I could not understand a manipilation...

    ΔeØ + k2eØ = 0

    ▼[▼ØeØ] + k2eØ = 0

    2ØeØ + (▼Ø)2eØ+k2eØ = 0

    I can not understand these manipilations... for a long time, I have searched the properties of laplecian but I could not find any propertiy in connection with laplecian...

    So, please explain how this property is working...

    thanks already for your helps...

    be well...
     
  2. jcsd
  3. Jun 25, 2009 #2

    Mute

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    Homework Helper

    It's an application of the rule

    [tex]\nabla \cdot (\phi \mathbf{F}) = \nabla\phi \cdot \mathbf{F} + \phi \nabla \cdot \mathbf{F}[/tex],

    where phi is a scalar function and F is a vector. For your case, the scalar function is [itex]\exp[\Theta][/itex] and the vector is [itex]\nabla \Theta[/itex].

    This comes from treating the laplacian as div grad: grad acts on exp(Theta) to give (grad Theta) exp[Theta] by the chain rule, and then the div acts on (grad Theta) exp[Theta].
     
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