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Derivation of Laplace in spherical co-ordinates

  1. Feb 13, 2014 #1
    I have been trying to derive the Laplace in spherical co ordinates.
    I have attached a file which has basic equations.
    I am trying to get the following equation.

    d(phi)/dx= -sin(phi)/(r sin (theta)).

    I have also attached the materials I am referring to.
    Can someone please help me derive the equation.

    Thank you.
     

    Attached Files:

  2. jcsd
  3. Feb 14, 2014 #2

    vanhees71

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    This is a pretty cumbersome way. The most easy is to use the action principle. The action
    [tex]A[\phi]=\int \mathrm{d}^3 \vec{x} [(\vec{\nabla} \phi)^2+f \phi][/tex]
    leads to the equation
    [tex]\Delta \phi=f.[/tex]
    Now you write the gradient in terms of spherical coordinates (which is easy to derive by your direct method)
    [tex]\vec{\nabla} \phi=\vec{e}_r \partial_r \phi+\frac{1}{r} \vec{e}_{\theta} \partial_{\theta} \phi + \frac{1}{r \sin \theta} \vec{e}_{\varphi} \partial_{\varphi} \phi.[/tex]
    The volume element is
    [tex]\mathrm{d}^3 {\vec{r}}=\mathrm{d} r \mathrm{d} \theta \mathrm{d} \varphi r^2 \sin \theta.[/tex]
    This you plug into the action integral and use the Euler-Lagrange equations to derive the field equation in terms of spherical coordinates. This leads to the Laplacian by identifying it with [itex]f[/tex].
     
  4. Feb 14, 2014 #3

    vela

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    Show us your attempt at deriving that equation.
     
  5. Feb 14, 2014 #4

    BvU

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    I second Vela. Vanhees is focusing on ##\Delta##. If you want a hint for your ## {d\phi \over dx} ## question: ##\tan \phi = {y \over x} ## is a good starting point.
     
  6. Feb 14, 2014 #5
    Thank you for your reply vanhees71. But I am not sure how you even got the equation.

    A[ϕ]=∫d3x→[(∇→ϕ)2+fϕ]

    I am a beginner trying to learn this derivation.

    Thank you BvU, your idea helped.

    Thank you all.
     
  7. Feb 14, 2014 #6
    Your attached references are just a straightforward transformation of coordinates that you learn how to do in a course on partial differential equations. Have you had a course that covers partial differential equations yet?

    Chet
     
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