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The volume of the n-ball () can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate r and angular coordinates φ 1, …, φ n − 1, where the domain of each φ except φ n − 1 is [0, π), and the domain of φ n − 1 is [0, 2 π). The spherical volume element is:
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n ...
When n > m the determinant and volume are zero. When n = m, this reduces to the standard theorem that the absolute value of the determinant of n n-dimensional vectors is the n-dimensional volume. The Gram determinant is also useful for computing the volume of the simplex formed by the vectors; its volume is Volume(parallelotope) / n!.
The determinant gives the signed n-dimensional volume of this parallelotope, () = (), and hence describes more generally the n-dimensional volume scaling factor of the linear transformation produced by A. [3] (The sign shows whether the transformation preserves or reverses orientation.)
Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: A half-space is a polyhedron defined by a single linear inequality, a 1 T x ≤ b 1. A hyperplane is a polyhedron defined by two inequalities, a 1 T x ≤ b 1 and a 1 T x ≥ b 1 (which is equivalent to -a 1 T x ≤ -b 1). A quadrant in the plane.
The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space. More formally, an (n + 1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ).
The n-dimensional volume of a n-dimensional hyperpyramid can be computed as follows: = Here V n denotes the n-dimensional volume of the hyperpyramid, A the (n – 1)-dimensional volume of the base and h the height, that is the distance between the apex and the (n – 1)-dimensional hyperplane containing the base A.
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 ...