1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is a covariant derivative

  1. Jul 23, 2014 #1
    Definition/Summary

    Covariant derivative, [itex]D[/itex], is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: [itex]D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0[/itex]

    The adjustment is made by a linear operator known both as the connection, [itex]\Gamma^i_{\ jk}[/itex], and as the Christoffel symbol, [itex]\{^{\ i\ }_{j\ k}\}[/itex].

    Covariant derivative of the metric ([itex]g_{ij}[/itex]) is zero.

    Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).

    A vector [itex]V^i[/itex] is parallel transported along a curve [itex]x^i(s)[/itex] with tangent [itex]T^i(s)[/itex] if its covariant directional derivative in the direction of that tangent is zero: [itex](T.D)V^i/ds\ =\ 0[/itex]

    Equations

    [tex]\frac{DV^i}{\partial x^j}\left(\text{also written}\ \frac{DV^i}{Dx^j}\ \text{or}\ \nabla_jV^i\ \text{or}\ V^i_{\ ;\,j}\right)\ =\ \frac{\partial V^i}{\partial x^j}\ -\ \Gamma^i_{\ jk}V^k[/tex]

    [tex]\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i[/tex]

    [tex]\frac{Dg^{ij}}{\partial x^k}\ =\ 0[/tex]

    [tex]\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}[/tex]

    [tex]\Gamma^i_{\ jk}\ =\ \{^{\ i\ }_{j\ k}\}\ =\ \frac{1}{2}\,g^{il}\left(\frac{\partial g_{jl}}{\partial x^k}\ +\ \frac{\partial g_{kl}}{\partial x^j}\ -\ \frac{\partial g_{jk}}{\partial x^l}\right)[/tex]

    Parallel transport of a vector [itex]V^i[/itex] along a curve [itex]x^i(s)[/itex]:

    [tex]\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j[/tex]

    Geodesic deviation equation:

    [tex]\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}[/tex]

    Extended explanation



    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted