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The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
Visualization of the whole observable universe.The inner blue ring indicates the approximate size of the Hubble volume. In cosmology, a Hubble volume (named for the astronomer Edwin Hubble) or Hubble sphere, subluminal sphere, causal sphere and sphere of causality is a spherical region of the observable universe surrounding an observer beyond which objects recede from that observer at a rate ...
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2]
In coordinates, a 3-sphere with center (C 0, C 1, C 2, C 3) and radius r is the set of all points (x 0, x 1, x 2, x 3) in real, 4-dimensional space (R 4) such that = = + + + =. ...
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
Volume is a measure of regions in three-dimensional space. [1] It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch).
The van der Waals radius, r w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms were not simply points and to demonstrate the physical consequences of their size through the van der Waals ...
We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension , we can apply the divergence theorem, i.e. =, and substitute for the volume integral of the divergence with the values of () evaluated at the cell surface (edges / and + /) of the finite volume as follows: