By Otto Forster

ISBN-10: 3834805750

ISBN-13: 9783834805751

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N} Man konstruiere einen Hom¨oomorphismus von W auf die Einheitskugel B1 (0) = {x ∈ Rn : x < 1}. 3. Man zeige, dass der Vektorraum C [a, b] aller stetigen Funktionen f : [a, b] → R auf dem kompakten Intervall [a, b] ⊂ R mit der Supremumsnorm f := sup{| f (x)| : x ∈ [a, b]} vollst¨andig ist. 4. Auf dem Vektorraum C 1 [a, b] aller einmal stetig differenzierbaren Funktionen f : [a, b] → R werde folgende Norm eingef¨uhrt: f C 1 := sup{| f (x)| + | f (x)| : x ∈ [a, b]}. a) Man zeige, dass C 1 [a, b] mit dieser Norm vollst¨andig ist.

Definition. Sei f : I → Rn eine stetig differenzierbare Kurve. Die Kurve heißt regul¨ar oder nicht-singul¨ar, falls f (t) = 0 f¨ur alle t ∈ I. Ein Parameterwert t ∈ I mit f (t) = 0 heißt singul¨ar. 6) Beispiel. Wir betrachten die Neilsche Parabel f : R −→ R2 , t → (t 2,t 3). F¨ur das Bild der Kurve gilt f (R) = {(x, y) ∈ R2 : x 0, y = ±x3/2 }, vgl. 4. Wegen f (t) = (2t, 3t 2) liegt f¨ur t = 0 in der Spitze der einzige singul¨are Punkt der Neilschen Parabel vor. 4 I. Differentialrechnung im Rn 40 Schnittwinkel Seien f : I1 → Rn und g : I2 → Rn zwei regul¨are Kurven.

Xn abh¨angt. 4) Sei n Die Funktion F ist aber auch an der Stelle x = 0 partiell differenzierbar mit ∂F F(hei ) − F(0) = 0, (0) = lim h→0 ∂xi h da F(hei ) = 0 f¨ur alle h ∈ R. Daraus folgt, dass F in ganz Rn partiell differenzierbar ist. F ist aber im Nullpunkt nicht stetig. Betrachten wir etwa die Werte von F auf der gegen 0 konvergierenden Punktfolge 1 1 , . , , k k k √ Es gilt r(ak ) = n/k, also ak := 1. (1/k)n F(ak ) = √ = n−n/2 ( n/k)n (unabh¨angig von k). Da limk→∞ F(ak ) = n−n/2 und F(0) = 0, ist F im Nullpunkt nicht stetig.

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Analysis 2 by Otto Forster

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