| _g__{μα} _g_^{αν} | = | δ^{ν}_{μ} | = 1
So that it follows (17) that | _g__{μν} | | _g_^{μν} | = 1.
_Invariant of volume._
We see first the transformation law for the determinant _g_ = | _g__{μν}
|. According to (11)
$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial
x_{\nu}}{\partial x_{\tau'}} g_{\mu u} | $$