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 formula_25. In coordinates, where formula_27 is the determinant of the metric tensor "g" written in the coordinate system. A simple example of a volume element can be explored by considering a two-dimensional surface embedded in "n"-dimensional Euclidean space. Such a volume element is sometimes called an "area element". Consider a subset formula_28 and a mapping function thus defining a surface embedded in formula_30. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form that allows one to