Surface Parametrizations and Their Jacobians
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.
You can drag the point in the domain. The red segment of length in the domain is mapped to the red tangent vector . Similarly, the blue segment of length in the domain is mapped to the blue tangent vector .
The concept illustrated in this Demonstration is key to understanding and defining surface area and surface integrals. See, for instance, the MathWorld link to surface integrals below, in which corresponds to and to .