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

Calculation: Formula for Laplacian/tr(Hess)

  1. May 22, 2013 #1
    Hi!

    I'm trying to understand the formula for the Laplace-Beltrami Operator on a Riemannian manifold.

    (http://en.wikipedia.org/wiki/List_o...vergence.2C_Laplace.E2.80.93Beltrami_operator)

    Specifically, how the determinant of the metric tensor comes into play when defining the the Laplace-Beltrami-Operator by trace(Hess f). I have found a computation by Peter Petersen but I don't understand one (probably very simple) step (See attachment).

    I would love to know how the determinant disappears in this step.

    Thank you in advance
     

    Attached Files:

    • del.jpg
      del.jpg
      File size:
      51.2 KB
      Views:
      69
  2. jcsd
  3. May 22, 2013 #2

    WannabeNewton

    User Avatar
    Science Advisor

    The following is a well known identity: [tex]\frac{1}{g}\frac{\partial g}{\partial x^{\mu}} = g^{\nu\alpha}\frac{\partial g_{\nu\alpha}}{\partial x^{\mu}}[/tex]

    where ##g = det(g_{\mu\nu})##

    Look up the formula(s) for the derivative of the determinant of a matrix and that should guide you through the derivation of the above identity. Cheers!
     
  4. May 22, 2013 #3
    Thank you!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculation: Formula for Laplacian/tr(Hess)
  1. Laplacian's theorem (Replies: 5)

Loading...