Given a parametrized surface, a little rectangle

in the domain of the parametrization

maps onto a segment of the surface. Approximate the area

of the segment by the area

of a parallelogram spanned by tangent vectors given by

and

. The area of the parallelogram is equal to a factor

times the area

of the rectangle. The local scaling factor

is sometimes called the Jacobian of the parametrization. As the base

and height

diminish (making the mesh finer), the ratio of the area of the parallelogram in the range to the corresponding area of the rectangle in the domain approaches the value of the Jacobian. Equivalently, the ratio of the areas of the segment of the surface and of the parallelogram approaches 1.