On surface area integration and the related mapping

摘要

This paper concerns the computation of a surface area given by a parametric representation on an arbitrary domain D. The basic approach is to construct an analytic mapping with nonzero Jacobian from a simply connected domain; say a unit square, to D. Then the integration used to determine the area can be easily performed on the unit square. For computational purposes the mapping is estimated by a standard relaxation method and an algorithm is developed to estimate the desired area by a finite sum. With sufficiently large sums the error of this estimation can be made arbitrarily small.