https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Semialgebraic_space 2026-08-01T04:30:30Z Revision history for this page on the wiki MediaWiki 1.47.0-wmf.7 https://en.formulasearchengine.com/w/index.php?title=Semialgebraic_space&diff=261610&oldid=prev 2014-05-05T18:48:46Z

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<p>added two examples of semi-algebraic functions (semi-algebraic functions should have their own wikipedia article)</p> <p><b>New page</b></p><div>In the [[mathematics|mathematical]] study of the [[differential geometry of surfaces]], the &#039;&#039;&#039;Bertrand–Diquet–Puiseux theorem&#039;&#039;&#039; <br /> expresses the [[Gaussian curvature]] of a surface in terms of the [[circumference]] of a [[geodesic]] circle, or the area of a geodesic disc. The theorem is named for [[Joseph Bertrand]], [[Victor Puiseux]], and C.F. Diquet.<br /> <br /> Let &#039;&#039;p&#039;&#039; be a point on a smooth surface &#039;&#039;M&#039;&#039;. The geodesic circle of radius &#039;&#039;r&#039;&#039; centered at &#039;&#039;p&#039;&#039; is the set of all points whose geodesic distance from &#039;&#039;p&#039;&#039; is equal to &#039;&#039;r&#039;&#039;. Let &#039;&#039;C&#039;&#039;(&#039;&#039;r&#039;&#039;) denote the circumference of this circle, and &#039;&#039;A&#039;&#039;(&#039;&#039;r&#039;&#039;) denote the area of the disc contained within the circle. The Bertrand–Diquet–Puiseux theorem asserts that<br /> :&lt;math&gt;K(p) = \lim_{r\to 0^+} 3\frac{2\pi r-C(r)}{\pi r^3} = \lim_{r\to 0^+}12\frac{\pi r^2-A(r)}{\pi r^4}.&lt;/math&gt;<br /> <br /> The theorem is closely related to the [[Gauss–Bonnet theorem]].<br /> <br /> ==References==<br /> *{{citation | last=Berger|first=Marcel | authorlink=Marcel Berger|title= A Panoramic View of Riemannian Geometry | publisher=Springer-Verlag | year=2004 | isbn=3-540-65317-1}}<br /> <br /> *{{citation|title=Démonstration d&#039;un théorème de Gauss|first1=J|last1=Bertrand|first2=C.F.|last2=Diquet|first3=V|last3=Puiseux|journal=Journal de Mathématiques|year=1848|volume=13|pages=80–90}}<br /> <br /> *{{citation|first=Michael|last=Spivak|authorlink=Michael Spivak|title=A comprehensive introduction to differential geometry, Volume II|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3}}<br /> <br /> {{DEFAULTSORT:Bertrand-Diquet-Puiseux theorem}}<br /> [[Category:Differential geometry of surfaces]]<br /> [[Category:Theorems in differential geometry]]<br /> <br /> <br /> {{differential-geometry-stub}}</div>

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