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In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system.
Liouville's equation can be used to prove the following classification results for surfaces: 7] A surface in the Euclidean 3-space with metric dl 2 = g(z, _)dzd _, and with constant scalar curvature K is locally isometric to: the sphere if K > 0; the Euclidean plane if K = 0; the Lobachevskian plane if K < 0.
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844 [1]), states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M ...
In complex analysis, see Liouville's theorem (complex analysis) There is also a related theorem on harmonic functions; In conformal mappings, see Liouville's theorem (conformal mappings) In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem; In linear differential equations, see Liouville's formula
The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers , and −1 if it is the product of an odd number of primes.
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge c {\displaystyle c} of its Virasoro symmetry algebra , but it is unitary only if
acceleration: meter per second squared (m/s 2) magnetic flux density also called the magnetic field density or magnetic induction tesla (T), or equivalently, weber per square meter (Wb/m 2) capacitance: farad (F) heat capacity: joule per kelvin (J⋅K −1) constant of integration: varied depending on context