Otto Forster's Analysis 2 PDF

By Otto Forster

ISBN-10: 3834805750

ISBN-13: 9783834805751

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L¨asst man nun (x, y) wobei nat¨urlich (ξ, η) und (ξ, ˜ η) ˜ → 0. d. Corollar. Sei U ⊂ Rn offen und f :U → R eine k-mal stetig partiell differenzierbare Funktion. Dann gilt Dik . . Di2 Di1 f = Diπ(k) . . Diπ(2) Diπ(1) f f¨ur alle i1 , i2 , . , ik ∈ {1, 2, . , n} und jede Permutation π der Zahlen 1, . , k. Der Beweis erfolgt durch vollst¨andige Induktion u¨ ber k unter Verwendung der Tatsache, dass sich jede Permutation aus Vertauschungen benachbarter Glieder zusammensetzen l¨asst. Schreibweise.

Seien K und L kompakte Teilmengen von Rn . Man zeige, dass dann auch die Menge K + L := {x + y : x ∈ K, y ∈ L} kompakt ist. 9. Seien I, J ⊂ R kompakte Intervalle und f : I × J → R eine stetige Funktion. Die Funktion F: I → R werde definiert durch F(x) := sup{ f (x, y) : y ∈ J}. Man zeige, dass F stetig ist. 36 § 4 Kurven im Rn ¨ Nach den bisherigen abstrakten Uberlegungen gehen wir jetzt wieder zur Untersuchung konkreter geometrischer Gebilde u¨ ber, n¨amlich von Kurven im Rn . Wir definieren Kurventangenten, Schnittwinkel von Kurven und behandeln den Begriff der Bogenl¨ange und ihre Berechnung.

Definition (Gradient). Sei U ⊂ Rn offen und f : U → R eine partiell differenzierbare Funktion. Dann heißt der Vektor ∂f ∂f (x), . , (x) grad f (x) = ∂x1 ∂xn der Gradient von f im Punkt x ∈ U . 2) definierte Funktion r gilt f¨ur x ∈ Rn x grad r = . r x Der Vektor r hat den Betrag 1 und die Richtung x. B. grad r = − 3 . 6) Seien f , g : U → R zwei partiell differenzierbare Funktionen. Dann gilt die Produktregel grad( f g) = g · grad f + f · grad g. Dies folgt aus der Produktregel f¨ur Funktionen einer Ver¨anderlichen, denn ∂ ∂f ∂g ( f g) = g+ f .

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Analysis 2 by Otto Forster

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