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Cristoffel Symbol of spherical coordinates

  1. Jun 30, 2014 #1
    I just derived the 3-D Cristoffel symbol of the 2nd kind for spherical coordinates. I don't think I made any careless mistakes, but once again, I just want to verify that I am correct and I can't find any place on line that will give me the components of the symbol so I can check myself.

    Here are the components that I derived: (I won't post the 0 components, nor will I post repeat components. By repeat components I mean: If I post [itex]\Gamma[/itex]212 then I already know that [itex]\Gamma[/itex]221 will be the same thing because you can switch around the bottom two indicies.)

    [itex]\Gamma[/itex]122 = -r

    [itex]\Gamma[/itex]133 = -rsin2(θ)

    [itex]\Gamma[/itex]212= 1/r

    [itex]\Gamma[/itex]233= -sin(θ)cos(θ)

    [itex]\Gamma[/itex]313= 1/r

    [itex]\Gamma[/itex]323 = cot(θ)

    I used the metric tensor and derivatives of metric tensors formula to derive these components.

    Can someone please look at these components that I derived and verify for me if I am right or not. I can provide the metric tensor that I used upon request if you wish to see further work.
     
  2. jcsd
  3. Jun 30, 2014 #2

    PeterDonis

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    You should provide the metric tensor, yes. Otherwise we won't know how to check your work.
     
  4. Jun 30, 2014 #3
    Here is my metric tensor gij :

    g11 = 1

    g22 = r2

    g33= r2sin2(θ)

    All other elements were 0. It was a 3 by 3 matrix.

    If you want my inverse version (the contravariant version that appears in the formula), then it is below:

    g11= 1

    g22= 1/r2

    g33 = 1/(r2sin2(θ))

    Once again all other elements were 0 and it was a 3 by 3 matrix.
     
  5. Jun 30, 2014 #4

    PeterDonis

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    Ah, ok. You mentioned "0 components" in the OP, but if you're just working in 3-dimensional Euclidean space, which you are with this metric, there are no "0" components. Your results look OK to me.
     
  6. Jun 30, 2014 #5
    Thank you very much. :smile:

    By the way, do you know of any places where I can just quickly check this stuff on line (for future reference)?

    Edit: Oh and by 0 components, I didn't mean 0th dimension like time or anything like that. I just meant elements that were 0. Sorry for the confusion.
     
  7. Jun 30, 2014 #6

    pervect

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    There are some programs that automate such calculations, some of them are free like Maxima.
     
  8. Jun 30, 2014 #7

    PeterDonis

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    Googling will sometimes turn up an online reference, but I don't know of any site that specifically tabulates this sort of thing in a systematic fashion for lots of different coordinate charts.

    To make these computations easier, I highly recommend learning how to use a symbolic math package. I use Maxima; other popular ones are Maple and MATLAB (which have the disadvantage of being a lot more expensive than Maxima, which is free :wink:). You can find out more about Maxima here:

    http://en.wikipedia.org/wiki/Maxima_(software [Broken])

    Maxima also has a package available called GRTensor that is specifically for computing things like Christoffel symbols.
     
    Last edited by a moderator: May 6, 2017
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