Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    Staff: Mentor

    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


    Staff: Mentor

    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


    User Avatar
    Staff Emeritus
    Science Advisor

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


    Staff: Mentor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook