Computing Christoffel symbols before studying curvature

[Terilien]

i'm having a hard time computing these so could people show me several examples to help me get a better feel for them before I move on to curvature?

 

[cristo]

One of the simplest examples would be to calculate the connection coefficients for the 3D Euclidean space using spherical polar coordinates. Here the line element is of the form $$ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$$. Could you try this one?

As an aside, have you studied and Lagrangian mechanics? If so, there is a method of obtaining the connection coefficients from the Euler-Lagrange equations which is sometimes less time consuming than using the definition involving the metric tensor.

 

[Terilien]

No but i know the euler lagrage equation.

 

[cristo]

Maybe it's easier to just use the definition of the symbols. Try plugging in the metric coefficients and see what you get for the gammas.

 

[Terilien]

Yes but could you still show me some examples, I'm not particularly comfortable with thses symbols.

 

[robphy]

http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html

 

[Terilien]

It's ok I'm fine now.

 

[cosmicstring1]

If you have Maple and GRTensor package you can calculate christoffel symbols for many metric files coming with the grtensor package and work them out yourself to exercise.