Question

why does the square root of the metric determinant is equal to the jacobian (im talking about curvilinear coordinates)

Answer 1

The metric determines the distances between points, and is usually written in terms of squares of lengths:

So for a change between coords {x_i} and {y_i} (sum over the i's and j's)

dx_i^2 = dy_i g_{ij} dy_j (I've assumed the x's are Cartesian)

where g_{ij} is the metric.

The Jacobian relates differential volume elements to one another. Volumes are products of single lengths(hence the square root), and since the sides of the volume elements aren't always perpendicular, you need the volume of a parallipiped, which gives the determinant.