Sphere Guide, Meaning , Facts, Information and Description
A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball). But in mathematics, a sphere is the boundary of a ball, and is therefore "hollow". This article deals with the mathematical concept of sphere.
Note that topologists and many geometers have different meanings for n-sphere, differing by one dimension. Thus, for example, 2-sphere means circle to many geometers but "ordinary" sphere to topologists. Some geometers use topologists' nomenclature, adding to the confusion.
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In three-dimensional Euclidean geometry, a sphere is the set of points in R3 which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.
In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the set of all points (x,y,z) such that
A sphere of any radius centered at the origin is described by the following differential equation:
The surface area of a sphere of radius r is:Geometry
Equations
The points on the sphere with radius r can be parametrized via
(see also trigonometric functions and spherical coordinates).
This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.
and its enclosed volume is:
The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes.
A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.
Spheres can be generalized to higher dimensions. For any natural number n, an n-sphere is the set of points in n-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number.
Spheres for n ≥ 4 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere.
In topology, an n-sphere is defined as the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere described above under Geometry, but perhaps lacking its metric. It is denoted Sn and is an n-manifold. A sphere need not be Differentiable manifolds|smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.
The Heine-Borel theorem helps to prove that an n-sphere is compact. You just have to prove that an n-sphere is closed and bounded:
The complement of Sn is Rn+1 \\ Sn. This is an open set. As the complement of an open set is closed Sn is closed.
Sn is also bounded. Therefor it is compact.
This is an Article on Sphere. Page Contains Information, Facts Details or Explanation Guide About Sphere Generalization to higher dimensions
However, see the note above about the ambiguity of n-sphere.See also
Topology
An n-sphere is an example of a compact n-manifold without boundary.See also
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