# Calculation: Formula for Laplacian/tr(Hess)

1. May 22, 2013

### Sajet

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.

#### Attached Files:

• ###### del.jpg
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2. May 22, 2013

### WannabeNewton

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

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!

3. May 22, 2013

Thank you!