Solid Angles in Arbitrary Dimensions
The solid angle subtended by the full surface of the unit n-sphere (in the geometer's sense) 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 2π rad for the 2D circumference and 4π sr for the 3D sphere. It also throws the slightly less obvious 2 for the 1D case, in which the origin-centered unit "sphere" is the set, which indeed has a measure of 2.
Read more about this topic: Solid Angle
Famous quotes containing the words solid, arbitrary and/or dimensions:
“The only solid and lasting peace between a man and his wife is, doubtless, a separation.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“You can’t be arbitrary about imposing your will when these people’s lives are at stake. Don’t you realize that? You speak as calmly as if you were saying “Pass the salt.” Aren’t you human? Are you made of ice?”
—Kurt Neumann (1906–1958)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)