 About 1,680 results  books.google.com 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
Chapter 7. 

 books.google.com 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 phasevolumepreserving condition (
Liouville's theorem) does not guarantee integrability for conservative systems in
general or ... 

 books.google.com ... 10, 62 sphere, 40 spherepreserving, 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,
Index 393. 

 books.google.com For dimension n=3, the transformations taking spheres into spheres account for
all anglepreserving transformations (Liouville's theorem). For n=2, the group of
transformations preserving angles is larger; however, even in this case the name
... 

 books.google.com 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 orientationpreserving and transforms every infinitesimal ball
into an ... 

 books.google.com 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
preserving. 

 books.google.com 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. 

 books.google.com 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 CauchyRiemann System is either constant or the restriction to fi of a Mobius
... They consider mappings of R which map spheres to spheres [10]. 

 books.google.com 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.
provides ... 

 books.google.com The simple timecentered 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 ... 

 