Read e-book online Klassische Mechanik PDF

By Walter Greiner

ISBN-10: 3817116985

ISBN-13: 9783817116980

ISBN-10: 3817117167

ISBN-13: 9783817117161

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2 (a) angesprochene Problem, in eine geschlossene Kurve Γ ⊂ 3 eine Graphenfl¨ ache kleinsten Fl¨ acheninhalts einzuspannen. Ê Ê Ist die Randkurve der Graph einer C 1 –differenzierbaren Funktion g : ∂Ω → (Ω ⊂ 2 ein Normalgebiet), so geht es um die Bestimmung der Minimumstellen mit des Fl¨ acheninhalts A : V → Ê Ê 1 + ∇v A(v) = 2 d2 x , V = C1g (Ω) . 2 die Euler–Gleichung UA div ∇u 1 + ∇u 2 = 0 in Ω . 3 (c). Fl¨ achen mit dieser Eigenschaft werden Minimalfl¨ achen genannt, auch wenn sie kein lokales Minimum von A auf V liefern.

Tp ) := F(u + sϕ + t1 ψ 1 + . . + tp ψ p ) definiert, da die Eintr¨ age nach (∗) zu V geh¨ oren. Ferner gilt gi (0, 0) = ci , ∂gi (0, 0) = δGi (u)ϕ , ∂s ∂f (0, 0) = δF(u)ϕ , ∂s ∂gi (0, 0) = δGi (u)ψ k , ∂tk ∂f (0, 0) = δF(u)ψ k f¨ ur 1 ≤ i, k ≤ p . ∂tk ∂gi (0, 0) = 0. Somit beschreiben die Gleichungen Nach Voraussetzung ist det ∂t k osungsmannigfaltiggi (s, t1 , . . , tp ) = ci (i = 1, . . , p) eine eindimensionale L¨ keit, auf der f an der Stelle (0, 0) ein lokales Minimum annimmt. Nach Bd.

Mit Hilfe der Supremumsnorm v C0 := max v(x) x∈Ω f¨ ur stetige Funktionen v : Ω → Ê + ∇u C0 u C1 := u C0 m . definieren wir auf C1 (Ω) die C1 –Norm 4 Mehrdimensionale Variationsprobleme 41 Ê (b) Der Gaußsche Integralsatz im 3 wurde f¨ ur Gaußsche Gebiete in Bd. 1, § 26 bewiesen. Wir formulieren hier kurz die Voraussetzungen f¨ ur eine allgemeinere onigsberger [130] § 10). Version im n (siehe Bd. 2, § 11 : 3 oder K¨ Ê Ê ar (r ≥ 1), Ein Randpunkt a ∈ ∂Ω eines Gebiets Ω ⊂ n heißt Cr –regul¨ wenn es eine Cr –Funktion ψ auf einer Umgebung U von a gibt mit nichtverschwindendem Gradienten und mit U ∩ Ω = {x ∈ U | ψ(x) < 0} , U \ Ω = {x ∈ U | ψ(x) ≥ 0} .

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Klassische Mechanik by Walter Greiner


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