GR Text: Zee's Formulas & Where I'm Going Wrong

[dm4b]

It seems like Zee lost a trace in his new GR text, but I am sure it is me confusing things.

First, he establishes:

$log\: det\: M = tr\: log\: M$

Then, differentiating:

$(det\: M)^{-1} \: \partial (det\: M)=\partial (tr\: log\: M)=tr(\partial\: log\: M)=tr(M^{-1}\partial M)$

Then he applies to the metric, giving:

$\frac{1}{\sqrt{-g}}\partial _{\nu }\sqrt{-g}=\frac{1}{2g}\partial _{\nu }g=\frac{1}{2}\partial _{\nu }log\: g$

where g is the determinant of the metric.

Sure seems like he lost a trace on the very last term. Where I am going wrong here?

 

[The_Duck]

In Zee's equation, $g$ means the determinant of the metric tensor. In more explicit notation, the steps are

$\frac{1}{\sqrt{-\det g}} \partial_\nu \sqrt{-\det g} = \frac{1}{2 \det g} \partial_\nu (\det g) = \frac{1}{2} \partial_\nu ({\rm tr} \log g) = \frac{1}{2} \partial_\nu (\log \det g)$

In what I've written here, $g$ is always a matrix and the determinant is kept explicit.