⇤ ⇤

�(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)