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The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is , where is the volume of the unit n-ball, the n-ball of radius 1.
In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume. [1] It is used throughout real analysis , in particular to define Lebesgue integration . Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable ; the measure of the Lebesgue-measurable set A is here denoted by λ ( A ).
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 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 ...
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract.It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1. Thus the volume of the n-ball ...
The content, i.e. the n-dimensional volume of this simplex, denoted by , can be expressed as a function of determinants of certain matrices, as follows: [4] [5] v n 2 = 1 ( n ! ) 2 2 n | 2 d 01 2 d 01 2 + d 02 2 − d 12 2 ⋯ d 01 2 + d 0 n 2 − d 1 n 2 d 01 2 + d 02 2 − d 12 2 2 d 02 2 ⋯ d 02 2 + d 0 n 2 − d 2 n 2 ⋮ ⋮ ⋱ ⋮ d 01 ...
In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold ...