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Del operator in spherical coordinates

  1. Mar 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Write the del operator in spherical coordinates?


    2. Relevant 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]


    3. The attempt at a solution
    I have no idea where to start... Please help, I'm going crazy with this...
     
  2. jcsd
  3. Mar 25, 2010 #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.
     
  4. Mar 27, 2010 #3

    vela

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