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23 12. h. 50) um zu b2 = r(a − e cos ϕ) und setzen Sie cos ϕ = xr ein. y P r r y r φ φ x e e e x−e F F b a 21 22 23 24 Als u ¨berraschende Konsequenz folgt, dass auch die Zeit bis zum tiefsten Punkt (eine Viertelperiode) unabh¨ angig von der Auslenkung ist. Christiaan Huygens, 1629–1695 (Den Haag). h. ein schwerer Massen” punkt gelangt auf dem Zykloidenweg in k¨ urzester Zeit von einem Punkt zu einem anderen, weiter unten liegenden; vgl. [10], [33]. ¨ Siehe auch die Ubungen zur Planetenbahn in Anhang B.

11 6. Fl¨ acheninhalt von Graphen: Gegeben sei eine offene Teilmenge U ⊂ R2 und eine C 1 -Funktion f : U → R. Die zugeh¨orige Graphenfl¨ ache ist die Abbildung X : U → R3 , X(x, y) = (x, y, f (x, y)). Man zeige, dass X eine Immersion ist und f¨ ur jede kompakte Teilmenge C ⊂ U gilt: 11 Anleitung: Es sei {c (u), n(u)} eine Orthonormalbasis von E(u). Wegen b ∈ E gilt (1) b = αc +βn f¨ ur gewisse α, β. Aus b ∈ E folgt (2) αc +βn = γc +δn f¨ ur ˜ existiert, muss dieselbe Bedingung f¨ gewisse γ, δ. Wenn X ur die ebene Kurve c˜ und das ebene Vektorfeld ˜b gelten.

30 2. Kurven 4. Kr¨ ummung bei beliebigen Parametern: F¨ ur eine regul¨are Kurve c : I → E2 wird die Kr¨ ummung wie folgt definiert: κ= det(c , c ) . 45) Zeigen Sie: a) Ist c nach Bogenl¨ ange parametrisiert, so ist κ die fr¨ uher definierte Kr¨ ummung, κ = c , Jc . 45) definiert, so gilt κ ˜ = κ ◦ α. Was gilt bei Orientierungsumkehrung (α < 0)? 5. h. die Kurve c : I → E2 , c(x) = ur [a, b] ⊂ I sowie die (x, f (x)). Bestimmen Sie die Bogenl¨ ange von c|[a,b] f¨ Kr¨ ummung von c in Abh¨ angigkeit von f und seinen Ableitungen f , f .

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A Comprehensive Introduction to Differential Geometry, VOL. 3, 2ND EDITION by Michael Spivak


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