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The sphere has the smallest total mean curvature among all convex solids with a given surface area. The mean curvature is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere. The sphere has constant mean curvature.
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.
The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ ( ℓ + 1) for some non-negative integer with ℓ ≥ | m | ; this ...
The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in [,): = (,) These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ().
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing.
Lines, L. (1965), Solid geometry: With Chapters on Space-lattices, Sphere-packs and Crystals, Dover. Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole.