Technische Universität München Christoph Niehoff Ferienkurs ...
Technische Universität München Christoph Niehoff Ferienkurs ...
Technische Universität München Christoph Niehoff Ferienkurs ...
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R = 1<br />
κ(t=0)<br />
<br />
γ(t) = (x(t), y(t)) T = t, t 2 T .<br />
<br />
<br />
κ(t) =<br />
˙x¨y − ¨x ˙y<br />
(t) =<br />
2<br />
.<br />
( ˙x 2 + ˙y 2 ) 3<br />
2<br />
(1 + 4t 2 ) 3<br />
2<br />
R = 1 1<br />
κ(t=0) = 2 <br />
<br />
˙γα(t) = αt α−1 , 1 .<br />
<br />
<br />
γα<br />
〈F (s), ds〉 =<br />
=<br />
=<br />
=<br />
=<br />
1<br />
0<br />
1<br />
0<br />
<br />
0<br />
1<br />
〈F (γα(t)) , ˙γα(t)〉 dt<br />
t 3 , t 3α , αt α−1 , 1 dt<br />
αt α+2 + t 3α dt<br />
α<br />
α + 3 tα+3 +<br />
1<br />
3α + 1 t3α+1<br />
1 0<br />
α<br />
α + 3 +<br />
1<br />
3α + 1 = 3α2 + 2α + 3<br />
3α2 + 10α + 3<br />
<br />
<br />
<br />
6 2−<br />
αx α , −y −α−3<br />
h(x, y) = 〈F (s), Gα(s)〉 = 2α 2 y 3 , x 3 ,<br />
<br />
= 2α 3 6 2−<br />
x α y 3 − 2α 2 x 3 y −α−3 .<br />
(h ◦ γα) (t) = 2α 3 (t α 6 2− α 3 2 α 3 −α−3 2 2α−3<br />
) (t) − 2α (t ) (t) = α (2α − 2) t<br />
<br />
<br />
γα<br />
〈F (s), Gα(s)〉 ds =<br />
=<br />
1<br />
0<br />
1<br />
0<br />
(h ◦ γα) (t) ˙γα(t) 2 dt<br />
α 2 (2α − 2) t 2α−3 1 + α 2 t 2α−2 dt<br />
ξ = 1 + α 2 t 2α−2 <br />
dt = α 2 (2α − 2) t 2α−3 −1 dξ <br />
<br />
γα<br />
〈F (s), Gα(s)〉 ds =<br />
<br />
1+α 2<br />
1<br />
= 2<br />
3<br />
ξ dξ<br />
1 3 <br />
2 2 + α − 1 .