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Del operator in spherical coordinates

  • Thread starter tirwit
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  • #1
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Homework Statement


Write the del operator in spherical coordinates?


Homework Equations


I wrote the spherical unit vectors:
[tex]\hat{r}[/tex]=sin[tex]\theta[/tex].cos[tex]\phi[/tex].[tex]\hat{x}[/tex]+sin[tex]\theta[/tex].sin[tex]\phi[/tex].[tex]\hat{y}[/tex]+cos[tex]\theta[/tex].[tex]\hat{z}[/tex]
[tex]\hat{\phi}[/tex]=-sin[tex]\phi[/tex].[tex]\hat{x}[/tex]+cos[tex]\phi[/tex].[tex]\hat{y}[/tex]
[tex]\hat{\theta}[/tex]=cos[tex]\phi[/tex].cos[tex]\theta[/tex].[tex]\hat{x}[/tex]+sin[tex]\phi[/tex]+cos[tex]\theta[/tex].[tex]\hat{y}[/tex]-sin[tex]\theta[/tex].[tex]\hat{z}[/tex]


The Attempt at a Solution


I have no idea where to start... Please help, I'm going crazy with this...
 

Answers and Replies

  • #2
vela
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In cartesian coordinates, you have

[tex]\nabla = \hat{x} \frac{\partial}{\partial x}+\hat{y} \frac{\partial}{\partial y}+\hat{z} \frac{\partial}{\partial z}[/tex]

Use the chain rule to change variables to [itex]r, \varphi, \vartheta[/itex]. For example, you can write

[tex]\frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \varphi}{\partial x}\frac{\partial}{\partial \varphi}+\frac{\partial \vartheta}{\partial x}\frac{\partial}{\partial \vartheta}[/tex]

Then express the cartesian unit vectors in terms of the spherical unit vectors. Plug everything in and simplify.
 
  • #3
vela
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