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In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold.The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation.
The Lie algebra of SO(3) is denoted by () and consists of all skew-symmetric 3 × 3 matrices. [7] This may be seen by differentiating the orthogonality condition , A T A = I , A ∈ SO(3) . [ nb 2 ] The Lie bracket of two elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} is, as for the Lie algebra of every matrix group, given by the ...
SU(2) → SO(3) are deformation retracts of the left and right groups, respectively, in the double covering SL(2, C) → SO + (1, 3). But the homogeneous space SO + (1, 3) / SO(3) is homeomorphic to hyperbolic 3-space H 3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H 3.
Thus as an irreducible representation of SO(3), H ℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ. More generally, the analogous statements hold in higher dimensions: the space H ℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-tensors.
Budgeting is more popular than ever. A 2022 Debt.com survey found that 86% of people track their monthly income and expenses, up from 80% in 2021 and 2020 and roughly 70% pre-pandemic. And in a ...
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Sulfur trioxide, SO 3, a chemical compound of sulfur and the anhydride of sulfuric acid; Sulfite, SO 2− 3, a chemical ion composed of sulfur and oxygen with a 2− charge; SO(3), the special orthogonal group in 3 dimensions; the rotations that can be given an object in 3-space; Star Ocean: Till the End of Time, the third main game in the Star ...
In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the ...