By P. Ciarlet

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Extra resources for An Intro. to Differential Geometry With Applns to Elasticity

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Bemerkung 1. Die Defmition der affmen Unabhiingigkeit hiingt nicht von der Reihenfolge der Punkte ab. Genauer gesagt: 1st (Po, ... , Pn) affm unabhiingig, so ist fUr jede Permutation a von {O, ... , n} auch (Pa(o), ... , Pa(n») affm unabhiingig. 22 1. Affine Geometrie Beweis. Da die lineare Unabhangigkeit nicht von der Reihenfolge der Vektoren abhangt, geniigt es zu zeigen, d~ fur jedes i E {O, ... , n} die Vektoren ---+ ---+ ---+- ---+ ~ PiPO,PiPh ···,PiPi-I,PiPi+I,'" ,PiPn linear unabhangig sind.

9), gilt dann fUr A:= B ·S-I, d~ g(x) = qo + A(x-po) fUr alle x E KD. Also folgt g(x) = b + Ax mit b:= qo-Apo Beispiel. FUr Po = (0, 1,0), und ist PI = (0,2,1), qo=(I,I), -8 A= ( 13 10 -16 ql=(-I,2), -12) 17 P3 = (-4,0,2)E IR 3 P2 = (1,3,1), q2=(I,-I), q3=(-I,-I)EIR 2 und Der Leser mage die erforderlichen Rechnungen zur Obung durchflihren und fUr einige x E IR 3 die Bildpunkte g(x) E IR2 bestimmen. 9. \t Fix(f) := {x E X: f(x) = x} die Menge der Fixpunkte von f. Bemerkung. Fix(f) eXist ein affiner Unterraum.

48). Fiir einen beJiebigen Punkt p der Schnittkurve gilt dann d(p, q) = d(p, F) und d(p, q') = d(p, F'), denn die Abstiinde von einem Punkt p au~erhalb einer Kugel zu den Beriihrpunkten der durch p gehenden Tangenten sind gieich. Also ist d(p, F) + d(p, F') = d(q, q') und somit unabhangig von p. S und S' he~en Dandelinsche Kugeln. E Brennpunktsgleichungen Gestiitzt auf die astronomischen Messungen von Tycho de Brahe entdeckte Johannes d~ sich die Planeten urn die Sonne nicht auf Kreisbalmen sondern auf elliptischen Balmen mit der Sonne im Brennpunkt bewegen.

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An Intro. to Differential Geometry With Applns to Elasticity by P. Ciarlet


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