# Derivation of Laplace in spherical co-ordinates

1. Feb 13, 2014

### darwined

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:

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• ###### The Laplacian Operator from Cartesian to Cylindrical to Spherical Coordinates.pdf
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2. Feb 14, 2014

### vanhees71

This is a pretty cumbersome way. The most easy is to use the action principle. The action
$$A[\phi]=\int \mathrm{d}^3 \vec{x} [(\vec{\nabla} \phi)^2+f \phi]$$
leads to the equation
$$\Delta \phi=f.$$
Now you write the gradient in terms of spherical coordinates (which is easy to derive by your direct method)
$$\vec{\nabla} \phi=\vec{e}_r \partial_r \phi+\frac{1}{r} \vec{e}_{\theta} \partial_{\theta} \phi + \frac{1}{r \sin \theta} \vec{e}_{\varphi} \partial_{\varphi} \phi.$$
The volume element is
$$\mathrm{d}^3 {\vec{r}}=\mathrm{d} r \mathrm{d} \theta \mathrm{d} \varphi r^2 \sin \theta.$$
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. Feb 14, 2014

### vela

Staff Emeritus
Show us your attempt at deriving that equation.

4. Feb 14, 2014

### BvU

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. Feb 14, 2014

### darwined

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. Feb 14, 2014

### Staff: Mentor

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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