Results 1 to 3 of 3

Math Help - covariant derivatives and christoffel symbols

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    249

    covariant derivatives and christoffel symbols

    0=\frac{\partial{g_{mn}}}{\partial{y^p}}+ \Gamma_{pm}^sg_{sn} + \Gamma_{pn}^rg_{mr}

    i am trying to verify this equality so here is what i did so far:

    \frac{\partial{g_{mn}}}{\partial{y^p}}+\frac{1}{2}  g^{sd}(\frac{\partial{g_{dm}}}{\partial{y^p}}+\fra  c{\partial{g_{dp}}}{\partial{y^m}}-\frac{\partial{g_{pm}}}{\partial{y^d}})g_{sn}+\fra  c{1}{2}g^{rz}(\frac{\partial{g_{zn}}}{\partial{y^p  }}+\frac{\partial{g_{zp}}}{\partial{y^n}}-\frac{\partial{g_{pn}}}{\partial{y^z}})g_{mr}

    then since g^{sd}g_{sn}=\delta^d_n i now have

    \frac{\partial{g_{mn}}}{\partial{y^p}}+\frac{1}{2}  \frac{\partial{g_{nm}}}{\partial{y^p}}+\frac{1}{2}  \frac{\partial{g_{np}}}{\partial{y^m}}-\frac{1}{2}\frac{\partial{g_{pm}}}{\partial{y^n}}+  \frac{1}{2}\frac{\partial{g_{mn}}}{\partial{y^p}}+  \frac{1}{2}\frac{\partial{g_{mp}}}{\partial{y^n}}-\frac{1}{2}\frac{\partial{g_{pn}}}{\partial{y^m}}

    and i seem to be stuck here. have what i've been doing so far correct and if so how may i continue. thanks.
    Last edited by oblixps; March 12th 2011 at 03:59 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2008
    Posts
    249
    oops i had a few typos in the christoffel symbols. now that they are fixed i had some nice cancellations.

    now i am left with:
    0=2\frac{\partial{g_{mn}}}{\partial{y^p}} and since the derivative of g is 0 and since it is a tensor and has a derivative of zero in cartesian coordinates, it must have the same for all coordinates and therefore the identity is 0 = 0. is this the correct reasoning? thanks.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2008
    Posts
    249
    i am still confused as i don't know how to get rid of the terms left over which is 2\frac{\partial{g_{mn}}}{\partial{y^p}} where g is the metric tensor.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Covariant & Contravariant differentiation
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 14th 2011, 11:25 AM
  2. covariant derivative with frame field
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: May 8th 2010, 02:31 PM
  3. problems in christoffel symbols
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 2nd 2010, 04:42 AM
  4. Riemann Christoffel, cyclic properties
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 29th 2009, 09:02 AM
  5. Christoffel-Darboux identity
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: October 12th 2007, 11:52 PM

Search Tags


/mathhelpforum @mathhelpforum