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Calculation: Formula for Laplacian/tr(Hess)

  1. May 22, 2013 #1

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


    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
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  2. jcsd
  3. May 22, 2013 #2


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    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!
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