Ad
related to: volume of a hypersphere formula physics example
Search results
Results From The WOW.Com Content Network
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
For example, the equation + + + = defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n. This hypersurface is also a smooth manifold, and is called a hypersphere or an (n – 1)-sphere.
It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold.
The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol S 5, and the equation for the 5-sphere, radius r, centre the origin is = {: ‖ ‖ =}. The volume of six-dimensional space bounded by this 5-sphere is
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol S 7, with formal definition for the 7-sphere with radius r of = {: ‖ ‖ =}.