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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 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.
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 mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, [1] Gheorghe Bratu [2] and Israel Gelfand. [3] The equation reads + = The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Emden–Chandrasekhar ...
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 ...
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Any Liouville number, in particular: Liouville's constant. Numbers with large irrationality measure , such as the Champernowne constant C 10 {\displaystyle C_{10}} (by Roth's theorem ). Numbers artificially constructed not to be algebraic periods .
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.