Surface of revolution never negative, the area is given by If the curve is described by the function , , then the integral becomes for revolution around the -axis, and for revolution around the "y"-axis (using ). These come from the above formula. For example, the spherical surface with unit radius is generated by the curve , , when ranges over . Its area is therefore For the case of the spherical curve with radius , rotated about the -axis A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic
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