About 1,680 results
|Inversion in spheres 83 §5.2. Conformal maps in Euclidean space 87 §5.3. |
Sphere preserving transformations 92 Chapter 6. The Classical Proof of
Liouville's Theorem 95 §6.1. Surface theory 95 §6.2. The classical proof 103
|The Lagrangian is L = (d0/dr)2 + (sin 0d0/dr)2 for a particle of unit mass on a |
sphere of unit radius. ... For/>2, the phase-volume-preserving condition (
Liouville's theorem) does not guarantee integrability for conservative systems in
general or ...
|... 10, 62 sphere, 40 sphere-preserving, 119, 122 spherical coordinates of a ball, |
350 cosine theorem, 48 Dirac operator, 227 ... 353 Gauß–Ostrogradski, 352
Green, 354 Hahn–Banach, 186 Heine–Borel, 76 identity, 180 Liouville, 146,
|For dimension n=3, the transformations taking spheres into spheres account for |
all angle-preserving transformations (Liouville's theorem). For n=2, the group of
transformations preserving angles is larger; however, even in this case the name
|Another example is provided by the Liouville theorem on conformal mappings in |
space. Visually, a mapping from a domain in n- dimensional Euclidean space is
conformal whenever it transforms each infinitesimal sphere into an infinitesimal
sphere. Möbius ... Roughly speaking, such a mapping is characterised by the
properties that it is orientation-preserving and transforms every infinitesimal ball
into an ...
|Here, he explicitly included inversion in spheres as a new example of a |
transformation that could be used in synthetic geometry in order to ... Moreover,
Liouville indicated how the transformation could yield elegant proofs of various
geometric theorems of Serret and Dupin. ... of the angles equals two right angles,
that is, the case in the transformed linear triangle, and the transformation is angle
|Spheres. In 1845 the young William Thomson (1824—1907) (later ennobled as |
Lord Kelvin) visited Paris where he had many ... In particular, Liouville
encouraged Thomson to write a paper comparing Faraday's new ideas on
electromagnetism with traditional French electrodynamics. ... and
correspondence about electrostatics led the former to his important theorem
about conformal mappings of space. ... He also pointed out that such
transformations are angle preserving or conformal.
|is sense preserving or sense reversing • Dlf(x) Df(x) = | J/(i)|2/nI for almost every x |
6 fi . Theorem 1.1. (Liouville Theorem) Every weak solution / : fi — ▻ Rn, n > 3, of
the Cauchy-Riemann System is either constant or the restriction to fi of a Mobius
... They consider mappings of R which map spheres to spheres .
|1.3 The Liouville theorem The first important theorem about conformal mappings |
in dimensions n > 3 was proved by ... 2 that / (orientation- preserving) is a
solution to the Cauchy -Riemann equations and therefore a holomorphic function
. ... In higher dimensions the group M6b(n) of all Mobius transformations of R '.
consisting of finite compositions of reflections in spheres and hyperplanes.
|The simple time-centered leapfrog integrator, coupled with forces derived from a |
potential, gives an area- preserving computation that has an exact Liouville
theorem. The physics is mapped directly to the computer, rather than first being ...