Διαφορική Γεωμετρία - Αρβανιτογεώργος - 2012
DESCRIPTION
diferential geometryTRANSCRIPT
⇤ ⇤
�(t) = (
45 cos t, 1� sin t,�3
5 cos t)
i k
ii N(t)
�(t) +
1kN(t) a
a1/k
� : R ! R3
�
000(t) = 0
f(x, y, z) = (x + y + z� 1)
2
c Sc = {(x, y, z) 2 R3: f(x, y, z) = c}
i Gauss K
X(u, v) = (u, v,
u2
2 +
v3
3 )
ii (u, v) K = 0, K > 0, K < 0;
p M
E = 2, F = 1, G = 1
e = 4, f = 1, g = 1 k1, k2Z1, Z2 M p
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ! ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ! ΤΟΜΕΑΣ ΘΕΩΡΗΤΙΚΩΝ ΜΑΘΗ-ΜΑΤΙΚΩΝ
!(t) = (t sin t, cos t, t) "# < t < #
i
ii t||!!(t)|| = 1;
iii (0, 1, 0)
! : I $ R3 !(I)
! !!(t) !!!(t)
N Gauss M M
p Sp . . . . . .Sp(v) = . . .
E, F,G e, f, g M
p . . .k1 k2 M p . . .
Gauss M p K = . . .
H = . . .. . . . . .
z = x2 " y2 (0, 0, 0)
Gauss
X(u, v) = (u2, eu cos v, eu sin v), "# < u < #, 0 < v < 2".
! !
E (xa)2 + (y
b)2 = 1 a, b > 0
!(t) = (x1(t), x2(t)) E
E (a, 0) (0, b)
! : I " R3 (0 # I)
" : I " R ! !(I)!
i
M
Gauss
ii d#p # : M1 " M2
M1,M2 p # M1
M z = x2 + 3xy $ 5y2
u1 = (1, 0, 0) u2 = (0, 1, 0)M p = (0, 0, 0)
!1, !2 : R " R3 !1(t) = (t, 0, t2) !2(t) =(0, t,$5t2) M p
p u1 u2
M
p # M
M p = (0, 0, 0)ii
Gauss Mp = (0, 0, 0)