Solid angle is much, much farther away. The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length
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