Surface area Surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods
Volume element compute the area of a set "B" lying on the surface by computing the integral Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is with index "i" running from 1 to "n", and "j" running from 1 to 2. The Euclidean metric in the "n"-dimensional space induces a metric formula_34 on the set "U", with matrix elements The determinant of the metric is given by For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2. Now consider a change of