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Laplacian operator in different coordinates

  1. Aug 12, 2007 #1
    how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?
     
  2. jcsd
  3. Aug 12, 2007 #2
    Cylindrical: Use the substitution [tex]r=\sqrt{x^2+y^2}[/tex] and [tex]\theta = \tan^{-1} \frac{y}{x}[/tex] assuming this is valid on this region.

    This leads to,
    [tex]\nabla^2 u = \frac{\partial ^2 u}{\partial r^2} +\frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2
    }\frac{\partial ^2 u}{\partial \theta ^2} + \frac{\partial ^2 u}{\partial z^2}=0[/tex]
    For the most part [tex]z[/tex] coordinate is not taken and that term vanished.

    Spherical: Using Spherical Coordinate substitutions:
    [tex]\nabla^2 u = \frac{1}{r^2} \left\{ \frac{\partial (r^2u_r)}{\partial r}+\csc^2 \theta \frac{\partial ^2 y}{\partial \theta^2}+\csc \theta \frac{\partial (\sin \phi u_{\phi})}{\partial \phi} \right\} = 0[/tex]
     
  4. Aug 13, 2007 #3
    in general (which is something you learn in vector analysis for physicists):
    [tex]\nabla^2=(\frac{h_3h_2}{h_1}\frac{\partial}{\partial u_1}\frac{\partial h_1}{\partial u_1},\frac{h_3h_1}{h_2}\frac{\partial}{\partial u_2}\frac{\partial h_2}{\partial u_2},\frac{h_1h_2}{h_3}\frac{\partial}{\partial u_3}\frac{\partial h_3}{\partial u_3})[/tex] or something like this.
    where:
    r=xi+yj+zk
    and h_i=|dr/du_i|
    i.e you take the norm of the vector.
     
    Last edited: Aug 13, 2007
  5. Aug 13, 2007 #4

    CompuChip

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    Cheat sheet (well not really cheating, unless you like deriving these things)
     
  6. Aug 13, 2007 #5

    dextercioby

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    Simply use the rules of change of variables in partial differentials. For example

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

    and then sub everyting in terms of the spherical coordinates. Then compute the 2-nd partial wrt to x and the same for y and z.
     
  7. Oct 19, 2009 #6
    Either I am confused at the moment, or it is not right. In general case it should be

    [tex]\nabla^2 \Phi= \frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial u_1} \left( \frac{h_2h_3}{h_1} \frac{\partial \Phi}{\partial u_1} \right) +

    \frac{\partial}{\partial u_2} \left( \frac{h_3h_1}{h_2}\frac{\partial \Phi}{\partial u_2} \right)

    +\frac{\partial}{\partial u_3} \left(\frac{h_1h_2}{h_3}\frac{\partial \Phi}{\partial u_3}\right)

    \right]
    [/tex]

    Source: Hobson, Mathematical methods for Physics and Engineering, pg. 374.
     
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