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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 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:
A subset of R n is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of R n has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure.
An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
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.
An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell: [2] V ≈ 4 π r 2 t , {\displaystyle V\approx 4\pi r^{2}t,}
Surface Class Surface Charts n-spheres: n-sphere S n: Hopf chart. Hyperspherical coordinates. Sphere S 2: Spherical coordinates. Stereographic chart Central projection chart Axial projection chart Mercator chart. 3-sphere S 3: Polar chart. Stereographic chart Mercator chart. Euclidean spaces: n-dimensional Euclidean space E n: Cartesian chart ...
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.