Derivation of Laplace in spherical co-ordinates

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  • #1
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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.
 

Attachments

  • Laplac.jpg
    Laplac.jpg
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  • derivation of the Laplacian from.pdf
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  • The Laplacian Operator from Cartesian to Cylindrical to Spherical Coordinates.pdf
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Answers and Replies

  • #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].
 
  • #3
vela
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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.
Show us your attempt at deriving that equation.
 
  • #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.
 
  • #5
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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.
 
  • #6
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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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