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Let be the surface area of the unit -sphere of radius embedded in -dimensional Euclidean space, and let be the volume of its interior, the unit -ball. The surface area of an arbitrary -sphere is proportional to the st power of the radius, and the volume of an arbitrary -ball is ...
The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 (r) is the surface area of an (n − 1)-sphere of radius r, then: = (). Applying this to the above integral gives the expression
The volume of the unit ball in Euclidean -space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit n {\displaystyle n} -ball, which we denote V n , {\displaystyle V_{n},} can be expressed by making use of the gamma function .
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 ...
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables is the radius, = is the circumference (the length of any one of its great circles), is the surface area,
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. [11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
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.
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space.More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense.