where is the metric and formula_50 denotes the partial derivative with respect to coordinate . The square-root of the (absolute value of the determinant of the) metric appears because the divergence must be written with the correct conception of the volume. In curvilinear coordinates, the basis vectors are no longer orthonormal; the determinant encodes the correct idea of volume in this case. It appears twice, here, once, so that the formula_51 can be transformed into "flat space" (where coordinates are actually orthonormal), and once again so that formula_50 is also transformed into "flat space", so that finally, the "ordinary" divergence can be written with the "ordinary" concept of volume in flat space ("i.e." unit volume, "i.e." one, "i.e." not written down). The square-root appears in the denominator, because the derivative transforms in the opposite way (contravariantly) to the vector (which is covariant). This idea of getting to a "flat coordinate system" where local computations can be done in a conventional way is called a vielbein. A different way to see this is to note that the divergence is the codifferential in disguise. That is, the divergence corresponds to the expression formula_53 with formula_54 the differential and formula_55 the Hodge star. The Hodge star, by its construction, causes the volume form to appear in all of the right places.