# Cristoffel Symbol of spherical coordinates

1. Jun 30, 2014

### space-time

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 $\Gamma$212 then I already know that $\Gamma$221 will be the same thing because you can switch around the bottom two indicies.)

$\Gamma$122 = -r

$\Gamma$133 = -rsin2(θ)

$\Gamma$212= 1/r

$\Gamma$233= -sin(θ)cos(θ)

$\Gamma$313= 1/r

$\Gamma$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. Jun 30, 2014

### Staff: Mentor

You should provide the metric tensor, yes. Otherwise we won't know how to check your work.

3. Jun 30, 2014

### space-time

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.

4. Jun 30, 2014

### 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.

5. Jun 30, 2014

### space-time

Thank you very much.

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.

6. Jun 30, 2014

### pervect

Staff Emeritus
There are some programs that automate such calculations, some of them are free like Maxima.

7. Jun 30, 2014

### 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 ). 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