Skew symmetric tensor

I am currently working through Tensor calculus and differential geometry by Prasun Nayak, however I am confused where something with skew symmetric tensors has came from.

In the last line I am confused why the determinant is 1. If the transformation was orthogonal I think this makes sense but there is no mention of the coordinate transformation being orthogonal. Can anyone shed light on this? Thank you

Solution 1:

Surely $g=|g_{ij}|$ depends on the coordinate system. So, either the transformation is metric-preserving, or else $g_{ij}$ will transform by $$\bar g_{k\ell} = \sum\frac{\partial u^i}{\partial\bar u^k}g_{ij}\frac{\partial u^j}{\partial\bar u^\ell}.$$ This means that $$\bar g = g\left(\frac{\partial(u^1,u^2)}{\partial(\bar u^1,\bar u^2)}\right)^2,$$ and $\sqrt{\bar g} = \sqrt{g}\frac{\partial(u^1,u^2)}{\partial(\bar u^1,\bar u^2)}$.

My vote is that the typesetter (proofreader) missed the bar under the square root.