Remember that the determinant is, up to constant multiples, the only dim(M)-form.
We have an abstract manifold M and a chart x.
A student (friend) is writing .
whereas the det should be a determinant and the basis vectors of T_p M induced by the chart.
I cann't see any sense in it. Maybe it is senseless. But maybe he uses some identifications (e.g. with some coefficients) or something like that.
What do you think?