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Partial derivative in Spherical Coordinates

  1. Sep 1, 2013 #1
    Is partial derivative of ##u(x,y,z)## equals to
    [tex]\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}[/tex]
    Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to
    [tex]\frac{\partial u}{\partial r}+\frac{\partial u}{\partial \theta}+\frac{\partial u}{\partial \phi}[/tex]

    Thanks
     
  2. jcsd
  3. Sep 1, 2013 #2

    Zondrina

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    I'm slightly confused about the question, but if you're just changing co-ordinates and taking partials it looks fine.
     
  4. Sep 1, 2013 #3
    Thanks for the reply. I just want to verify partial derivative in TRUE Spherical Coordinates system is [tex]\frac{\partial u}{\partial r}+\frac{\partial u}{\partial \theta}+\frac{\partial u}{\partial \phi}[/tex]

    Not the spherical amplitude representation of (x,y,z) where
    [tex]\vec r=\hat x x +\hat y y+\hat z z\;\hbox { to which}\; x=r\cos\phi\sin\theta,\;y=r\sin\phi\sin\theta, \;z=r\cos\theta[/tex]
    In the ordinary Calculus III class.

    Thanks
     
  5. Sep 1, 2013 #4
    You are just differentiating wrt different functions, any calculations done by either method should give you same end results.
     
  6. Sep 1, 2013 #5
    Thanks
     
  7. Sep 1, 2013 #6

    vela

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    You need to get your terminology straight if you want people to understand you. The expression you wrote above is for the [strike]divergence[/strike] directional derivative of u in the (1,1,1) direction times the square root of 3. Each term is a partial derivative. So the answer to your question is no.

    No.
     
    Last edited: Sep 1, 2013
  8. Sep 1, 2013 #7

    vanhees71

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    This expression is not a divergence. The divergence is defined for a vector field. In Cartesian coordinates it reads
    [tex]\vec{\nabla} \cdot \vec{V}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}.
    [/tex]
     
  9. Sep 1, 2013 #8

    vela

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    D'oh!
     
  10. Sep 1, 2013 #9
    Yes, I know Divergence, Gradient and Laplace/Poisson equation in Spherical. Just the simple partial derivative.
     
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