Download PDF by Sagan H.: Advanced calculus

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A (r, cp) + sm r x r x2x1 2 &u cos 2 cp 0 2 (p(r, X1 a2u (p(r,cp )) , . 2 cpa +sm 2 x2 &u cp)) + 2 cos cp sin cp-a-(p(r, cp)) x 1x 2 §6. l(r, cp) v

JEM; Danun n n A=UAicU i=O und IMil kompakt. < U Ui= U i=OjEM; ui n iEUM; oo fiir aile i E {0, ... , n }, ist auch tQo Mil < oo. Also ist A 39 §3. Kompaktheit Bemerkung. a. nicht wieder kompakt zu sein braucht. Beispiel: X = lR, An := [0, 1 - 1/n], n ;: : : 1. Dann ist An fiir jedes n ;: : : 1 kompakt, aber 00 00 n=1 n=1 U An= U [o, 1- ~] = [0, 1[ ist nicht kompakt. Aufgabe 3D. n beliebig und zu jeder Folge (xi )iEN von Punkten Xi E A gebe es eine Teilfolge (xik)kEN• die gegen einen Punkt a E A konvergiert.

A( -x) = -a(x), b( -x) = b(x) --+ lR zwei stetige fiir aile x E I. Man zeige: Die Differentialgleichung y" + a(x)y' + b(x)y = 0 besitzt ein Fundamentalsystem von LOsungen, das aus einer geraden und einer ungeraden Funktion besteht. 26 Aufgaben §13. Lineare Ditlerentialgleichungen mit konstanten Koeffizienten Aufgabe 13 A. Man bestimme ein reelles Fundamentalsystem von Losungen fiir die folgenden Differentialgleichungen: a) y" - 4y' + 4y = 0, b) y"'- 2y"- 5y' + 6y = 0, c) y"' - 2y" + 2y' - y = 0, d) y"' - y = 0, e) y< 4 > + y = 0, f) y( 8 ) + 4y(6 ) + 6y(4 ) + 4y" + y = 0.

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Advanced calculus by Sagan H.

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