url - geometria diferencial
TRANSCRIPT
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Batista, Ricardo Alexandre Tópicos de geometria diferencial / Ricardo AlexandreBatista. - Rio Claro : [s.n.], 2011 91 f. : il., figs.
Dissertação (mestrado) - Universidade Estadual Paulista,Instituto de Geociências e Ciências Exatas Orientador: João Peres Vieira
1. Geometria diferencial. 2. Aplicação de Gauss. 3.Curvatura gaussiana. 4. Superfícies mínimas. 5. TeoremaEgregium de Gauss. 6. Teorema de Gauss Bonnet. I. Título.
516.36B333t
Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESPCampus de Rio Claro/SP
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α : I → R3 I =]a, b[ R R3
α : I → R3
α′(t) ̸= 0
t ∈ I
α : I → R3
t0, t1 ∈ I t0 ≤ t1 α t0 t1
t1 − t0 t1t0
∥α′(t)∥ dt = t1 − t0.
α : I → R3
s ∈ I ∥α′′(s)∥ = kα(s) α s
kα(s) ̸= 0 α′′(s) = kα(s)nα(s)
nα(s) α′′(s)
s
tα(s) = α′(s)
α
s
t′α(s) = kα(s)nα(s)
bα(s) = tα(s) ∧ nα(s) s
α : I → R3
s
α′′(s) ̸= 0, s
∈ I
τ α(s) b
′
α
(s) = τ α(s)nα(s)
α
s
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k(s), s ∈ I ⊂ R α(s)
s
k(s)
α(s)
α(s0) = p0 α′(s0) = v0 v0
R
2
α(s)
β (s)
L
T
R
2
α(s) = (L ◦ T )(β (s))
R3
S ⊂ R3 p ∈ S
V
p R
3
χ : U → V ∩ S
U
R
3
V ∩ S
χ
χ
q ∈U
dχq : R2→R3
χ
p
V ∩ S
p
S
p
∈ S
S
p
S
p ∈ S
α′(t0) α ]a, b[ → S ⊂ R3 α(t0) = p
V
W
T : V → W
V
T
T
T = {
v ∈
V :T (v) = 0}
T
T = T (V ) = {T (v) : v ∈ V } ⊂ W
T
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χ : U ⊂ R2 → S
S
q ∈ U
dχq(R2) ⊂ R3
2
p = χ(q )
S
dχq : R2
→R
3
dχq R2
dχq(R2)
dχq(R2)
S
p
T pS dχq(R
2)
p = χ(q )
χ : U ⊂R2→S
S
q ∈ U
T pS χu(q ) χv(q ) p = χ(q )
dχq : R2 → R3
(a, b) →
∂x∂u
(q ) ∂x∂v
(q )∂y∂u
(q ) ∂y∂v
(q )
∂z∂u (q ) ∂z∂v (q )
a
b
χu(q ) =
∂x∂u
(q ), ∂y∂u
(q ), ∂z∂u
(q )
χv(q ) =
∂x∂v
(q ), ∂y∂v
(q ), ∂z∂v
(q )
dχq
{e1, e2} R2 χu(q ) = dχq(e1) χv(q ) = dχq(e2)
dχq(R2)
{χu(q ), χv(q )} T pS = dχq(R2)
χ : U ⊂ R2 → S S
p ∈ χ(U ) ⊂ S N ( p) =
χu∧χv∥χu∧χv∥ (q ) χ(q ) = p χu ∧ χv χu χv
N : χ(U ) → R3
p ∈ χ(U )
N ( p)
V ⊂ S S N : V → R3
v ∈ V
v
N
V
N
S
N
S
T pS p ∈ S
{v, w
} ⊂ T
pS
⟨v ∧ w, N ⟩ > 0
T pS
T pS
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S 1 S 2 f : S 1
→ S 2
p ∈ S 1 χ χ̄ S 1 S 2
χ : U → S 1, χ̄ : Ū → S 2
p ∈ χ(U )
f ( p) ∈ χ̄(Ū )
f (χ(U )) ⊂ χ̄(Ū )
h = χ̄−1 ◦ f ◦ χ : U → Ū
f
S 1 S 2 f
f −1
S 1 S 2 f : S 1 → S 2
f
p
S 1
w1
w2 ∈ T pS 1
< w1, w2 >=< df p(w1), df p(w2) > .
f : S 1 → S 2 S 1 S 2
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S
p
S
S
N
S
N
S ⊂ R3
N
N :
S → R3
S 2 = {(x , y , z ) ∈ R3 : x2 + y2 + z 2 = 1}
N : S → S 2 Gauss S
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dN p
N
p ∈ S
T pS T N ( p)S 2
T pS T N ( p)S 2
dN p
T pS
dN p : T pS → T pS
α(t)
S
α(0) = p
N oα(t) =
N (t)
S
2
N
α(t)
N ′(0) = dN p(α′(0)) T pS
N
α(t)
t = 0
dN p
N
N ( p)
p
curvatura
dN p : T pS → T pS
dN p dN p
< dN p(v), w >=< v, dN p(w) > {v, w} T pS
χ(u, v)
S
p {χu, χv} T pS
α(t) = χ(u(t), v(t))
S
α(0) = p
dN p(χu)u′(0) + dN p(χv)v′(0) = dN p(χuu′(0) + χvv′(0)) = dN p(α′(0)) = ddtN (u(t), v(t))
|t=0 = N uu′(0) + N vv′(0)
dN p(χu) = N u dN p(χv) = N v dN p
< N u, χv >=< χu, N v >
N =
χu∧χv∥χu∧χv∥
< N, χu >= 0 ⇒< N v, χu > + < N, χuv >= 0
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< N, χv >= 0 ⇒< N u, χv > + < N, χvu >= 0
(3.2) (3.1)
< N v
, χu
> −
< N u
, χv
>= 0
< χu
, N v
>=< N v
, χu
>=< N u
, χv
>
dN p dN p
Q
T pS Q(v) =< dN p(v), v > v ∈ T pS
∏ p T pS
∏ p(v) = − < dN p(v), v >
S
p
C
p ∈ S k
C
p
cosθ =< n, N >
n
C
N
kn = k cos θ C ⊂ S
C ⊂ S α(s) s
C
α(0) = p
N (s)
N
α(s)
< N (s), α′(s) >= 0
< N ′(s), α′(s) > + < N (s), α′′(s) >= 0
< N (s), α′′(s) > − < N ′(s), α′(s) >
∀s
∏ p(α
′(0)) = − < dN p(α′(0)), α′(0) >= − < N ′(0), α′(0) >=< N (0), α′′(0) >=< N (0), k(0)n(0) >= k(0) < N (0), n(0) >= k(0)cos θ
θ
n(0)
N (0)
∏ p(α
′(0)) = kn( p)
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∏ p
v ∈ T pS p v
A : V → V
{e1, e2} V A(e1) = λ1e1 A(e2) = λ2e2 e1 e2
λ1 λ2 A
{e1, e2} A
λ1 λ2 λ1 ≥ λ2
Q(v) =< A v , v >
V
v ∈ V v = xe1 + ye2
x2 + y2 = 1
Q(v) =< Av, v >=< xA(e1) + yA(e2), xe1 + ye2 >=< xλ1e1 + yλ2e2, xe1 + ye2 >=
λ1x2 + λ2y
2
λ1 ≥ λ2
Q(v) = λ1x2 + λ2y
2 ≥ λ2(x2 + y2) = λ2
Q(v) = λ1x2 + λ2y
2 ≤ λ1(x2 + y2) = λ1
λ2
≤ Q(v)
≤ λ1 v V
Q(1, 0) = λ1 ≥ Q(0, 1) = λ2 λ2 λ1
Q(v)
A = −dN p
p ∈ S
{e1, e2} T pS −dN p(e1) =k1e1 −dN p(e2) = k2e2
k1 k2 (k1 ≥ k2)
∏ p T pS
p
k1
k2 p
e1 e2 p
T : V → V
dN p
−k1 00 −k2
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dN p (−k1)(−k2) = k1k2
dN p −(k1 + k2)
p ∈ S
dN p : T pS → T pS
dN P K S p
dN p
k1 k2 K = k1k2 H =
k1+k22
χ : U ⊂ R2 → S
N
S
χ(U )
N ( p) = χu∧χv∥χu∧χv∥(q ) p = χ(q ) ∈ χ(U )
χ(u, v)
p
S
α(t) = χ(u(t), v(t))
S
α(0) = χ(q ) = p
q = (u(0), v(0))
α(t)
p
α′(0) = χu(q )u′(0) + χv(q )v′(0) dN p(α′(0)) =
N ′(0) = N u(q )u′(0)+ N v(q )v′(0)
N (t) = N (χ(u(t), v(t)))
N (u(t), v(t))
< N , N >= 1
< N u, N > + < N, N u >= 0 < N u, N >=
0
< N v, N >= 0 N u N v T pS
N u = a11χu + a21χv
N v = a12χu + a22χv
dN p(α′(0)) = N u(q )u′(0) + N v(q )v′(0) = (a11χu(q ) + a21χv(q ))u′(0) + (a12χu(q ) +
a22χv(q ))v′(0) = (a11u′(0) + a12v′(0))χu(q ) + (a21u′(0) + a22v′(0))χv(q )
dN p
u′(0)
v′(0)
=
a11 a12
a21 a22
u′(0)
v′(0)
{χu(q ), χv(q )
} dN p (aij) i, j = 1, 2
{χu(q ), χv(q )}
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∏ p(α
′(0)) = − < dN p(α′(0)), α′(0) >= − < N u(q )u′(0) + N v(q )v′(0), χu(q )u′(0) + χv(q )v′(0) >
= −u′(0)2
< N u(q ), χu(q ) > −u′(0)v′(0) < N u(q ), χv(q ) >−v′(0)u′(0) < N v(q ), χu(q ) > −v′(0)2 < N v(q ), χv(q ) >
< N, χu >= 0 =< N, χv >
• < N u, χu > + < N, χuu >= 0
< N, χuu >= − < N u, χu >
• < N u, χv > + < N, χvu >= 0
< N, χuv >=< N, χvu >= − < N u, χv >
• < N v, χu > + < N, χuv >= 0
< N, χuv >= − < N v, χu >
• < N v, χv > + < N, χvv >= 0
< N, χvv >=
− < N v, χv >
∏ p(α
′(0)) = < N ( p), χuu(q ) > u′(0)2 + 2 < N ( p), χuv(q ) > u′(0)v′(0)
+ < N ( p), χvv(q ) > v′(0)2
p = χ(q )
< N ( p), χuu(q ) >= e(q )
< N ( p), χuv(q ) >= f (q )
< N ( p), χvv(q ) >= g(q )
∏ p(α
′(0)) = e(q )u′(0)2 + 2f (q )u′(0)v′(0) + g(q )v′(0)2
dN ( p
)
det(
aij)
N u( p) = a11χu(q ) + a21χv(q )
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N v( p) = a12χu(q ) + a22χv(q )
E (q ) =< χu(q ), χu(q ) >
F (q ) =< χu(q ), χv(q ) >
G(q ) =< χv(q ), χv(q ) >
−e(q ) = < N
u( p), χ
u(q ) >
= < a11χu(q ) + a21χv(q ), χu(q ) >
= a11 < χu(q ), χu(q ) > +a21 < χv(q ), χu(q ) >,
−e(q ) = a11E (q ) + a21F (q )
−f (q ) = < N u( p), χv(q ) >= < a11χu(q ) + a21χv(q ), χv(q ) >
= a11 < χu(q ), χv(q ) > +a21 < χv(q ), χv(q ) >,
−f (q ) = a11F (q ) + a21G(q )
−f (q ) = < N v( p), χu(q ) >= < a12χu(q ) + a22χv(q ), χu(q ) >
= a12 < χu(q ), χu(q ) > +a22 < χv(q ), χu(q ) >,
−f (q ) = a
12E (q ) + a
22F (q )
−g(q ) = < N v( p), χv(q ) >= < a12χu(q ) + a22χv(q ), χv(q ) >
= a12 < χu(q ), χv(q ) > +a22 < χv(q ), χv(q ) >,
−g(q ) = a12F (q ) + a22G(q )
(3.9)
(3.12)
a11 a21
a12 a22
E (q ) F (q )
F (q ) G(q )
= −
e(q ) f (q )
f (q ) g(q )
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∥χu(q ) ∧ χv(q )∥2 = ∥χu(q )∥2 ∥χv(q )∥2 sen2θ θ χu(q ) χv(q )
∥χu(q ) ∧ χv(q )∥2 = ∥χu(q )∥2 ∥χv(q )∥2 (1 − cos2θ)= ∥χu(q )∥
2
∥χv(q )∥2
− (∥χu(q )∥ ∥χv(q )∥ cosθ)2
= ∥χu(q )∥2 ∥χv(q )∥2 − < χu(q ), χv(q ) >2
∥χu(q ) ∧ χv(q )∥2 = E (q )G(q ) − F 2(q ) E (q )G(q ) − F 2(q ) > 0
E (q ) F (q )
F (q ) G(q )
a11 a21a12 a22
= − e(q ) f (q )
f (q ) g(q )
E (q ) F (q )F (q ) G(q )
−1
( )−1
( )
det
a11 a12
a21 a22
= det
a11 a21
a12 a22
= det e(q ) f (q )
f (q ) g(q ) 1
E (q ) F (q )F (q ) G(q )
=
e(q )g(q ) − f 2(q )E (q )G(q ) − F 2(q )
K (q ) = e(q )g(q ) − f 2(q )E (q )G(q ) − F 2(q )
a11 a22 (3.13)
a11 a21
a12 a22
= −
e(q ) f (q )
f (q ) g(q )
E (q ) F (q )
F (q ) G(q )
−1
E (q ) F (q )
F (q ) G(q )
−1=
1
E (q )G(q ) − F 2(q )
G(q ) −F (q )−F (q ) E (q )
t,
( )t
( )
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a11 a21
a12 a22
= − 1
E (q )G(q ) − F 2(q )
e(q ) f (q )
f (q ) g(q )
G(q ) −F (q )−F (q ) E (q )
a11 = f (q )F (q ) − e(q )G(q )
E (q )G(q ) − F 2(q )
a22 = f (q )F (q ) − g(q )E (q )
E (q )G(q ) − F 2(q )
−k1 −k2 dN p k1 k2
dN p(v) = −λv = −λI (v) v ∈ T pS v ̸= 0 I
(dN p + λI )(v) = 0 v ∈ T pS v ̸= 0
ker(dN p + λI ) ̸= 0 dN p + λI
det
a11 + λ a12
a21 a22 + λ
= 0,
(a11 + λ)(a22 + λ) − a21a12 = 0
λ2 + (a11 + a22)λ + a11a22 − a21a12 = 0
λ2 + (a11 + a22)λ + K (q ) = 0
k1 k2
H (q ) = k1 + k2
2 =
−(a11 + a22)2
= e(q )G(q ) − 2f (q )F (q ) + g(q )E (q )
2(E (q )G(q ) − F 2(q ))
H (q )
a11 + a22 = −2H (q ) (3.18)
λ2 − 2H (q )λ + K (q ) = 0
H 2(q ) − K (q ) = (k1 − k2)2
4 ≥ 0
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λ = H (q ) ±√
H 2(q ) − K (q )
H (q ) +√
H 2(q ) − K (q ) H (q ) −√ H 2(q ) − K (q )
< u∧v,w >= (u , v , w)
u
v
w R
3
< u ∧ v,w >
u ∧ v
w
u ∧ v
u
v
(u , v , w)
3 × 3
u
v
w �⃗�
i,⃗ j,⃗ k
R3
χ(u, v) = (x(u, v), y(u, v), z (u, v))
K (u, v) = a2 > 0
χ̄(u, v) = (ax(u, v), ay(u, v), az (u, v))
K̄ (u, v) = a−2K (u, v) = 1
χ̄(u, v)
K̄ (u, v) = 1
χ(u, v)
K (u, v) = a2 > 0
K̄ (q ) = ē(q )ḡ(q ) − f̄ 2(q )Ē (q ) Ḡ(q ) − F̄ 2(q )
Ē (q ) =< χ̄u, χ̄u >=< aχu, aχu >= a2 < χu, χu >= a
2E (q )
F̄ (q ) =< χ̄u, χ̄v >=< aχu, aχv >= a2 < χu, χv >= a
2F (q )
Ḡ(q ) =< χ̄v, χ̄v >=< aχv, aχv >= a2 < χv, χv >= a
2G(q )
ē(q ) =< N̄ ( p), χ̄uu >
f̄ (q ) =< N̄ ( p), χ̄uv >
ḡ(q ) =< N̄ ( p), χ̄vv >
N̄ ( p) = χ̄u ∧ χ̄v∥χ̄u ∧ χ̄v∥(q )
=
aχu
∧aχv
∥aχu ∧ aχv∥(q )=
χu ∧ χv∥χu ∧ χv∥(q ) = N ( p)
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ē(q ) =< N̄ ( p), χ̄uu >=< N ( p), aχuu >= ae(q )
f̄ (q ) =< N̄ ( p), χ̄uv >=< N ( p), aχuv >= af (q )
ḡ(q ) =< N̄ ( p), χ̄vv >=< N ( p), aχvv >= ag(q )
K̄ (u, v) = ē(q )ḡ(q ) − f̄ 2(q )Ē (q ) Ḡ(q ) − F̄ 2(q )
= ae(q )ag(q ) − (af (q ))2a2E (q )a2G(q ) − (a2F (q ))2
= a2e(q )g(q ) − a2f 2(q )a4E (q )G(q ) − a4F 2(q )
=
a2(e(q )g(q )
−f 2(q ))
a4(E (q )G(q ) − F 2(q ))= a−2K (u, v)
K (u, v) = a2 > 0
K̄ (u, v) = 1
K̄ (u, v) = 1
K (u, v) = a2 > 0
χ(u, v) = (f (u)cos v, f (u)sen v, g(u))
f (u) > 0 f ′(u)2 + g ′(u)2 = 1
u
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E (u, v) = < χu(u, v), χu(u, v) >
=
= f ′(u)2
cos
2
v + f ′(u)2
sen
2
v + g′(u)2
= f ′(u)2 + g′(u)2
= 1
F (u, v) = < χu(u, v), χv(u, v) >
= = −f ′(u)f (u)cos v sen v + f ′(u)f (u)cos v sen v= 0
G(u, v) = < χv(u, v), χv(u, v) >
= = f (u)2 sen2 v + f (u)2 cos2 v
= f (u)2
e(q ) = < N (u, v), χuu(u, v) >
= < χu(u, v) ∧ χv(u, v)∥χu(u, v) ∧ χv(u, v)∥ , χuu(u, v) >
= (χu(u, v), χv(u, v), χuu(u, v))∥χu(u, v) ∧ χv(u, v)∥
χuu(u, v) = (f ′′(u)cos v, f ′′(u)sen v, g′′(u))
(χu(u, v), χv(u, v), χuu(u, v)) =
f ′(u)cos v −f (u)sen v f ′′(u)cos vf ′(u)sen v f (u)cos v f ′′(u)sen v
g′(u) 0 g′′(u)
= f ′(u)f (u)g′′(u)cos2 v − f ′′(u)f (u)g′(u)sen2 v
−f ′′(u)f (u)g′(u)cos2 v + f ′(u)f (u)g′′(u)sen2 v
= f ′(u)f (u)g′′(u) − f ′′(u)f (u)g′(u)= f (u)(f ′(u)g′′(u) − f ′′(u)g′(u))
χu(u, v) ∧ χv(u, v) =
⃗ i ⃗ j ⃗ k
f ′(u)cos v f ′(u)sen v g′(u)
−f (u)sen(v) f (u)cos v 0
= (−
f (u)g′(u)cos v,
−f (u)g′(u)sen v, f ′(u)f (u)cos2 v + f ′(u)f (u)sen2 v)
= (−f (u)g′(u)cos v, −f (u)g′(u)sen v, f ′(u)f (u))
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∥χu(u, v) ∧ χv(u, v)∥ =√
f (u)2g′(u)2 cos2 v + f (u)2g′(u)2 sen2 v + f ′(u)2f (u)2
=
√ f (u)2g′(u)2 + f ′(u)2f (u)2
= √ f (u)2(g′(u)2 + f ′(u)2)= f (u)
e(u, v) = f (u)(f ′(u)g′′(u) − f ′′(u)g′(u))
f (u)
= f ′(u)g′′(u) − f ′′(u)g′(u)f (u, v) = < N (u, v), χuv(u, v) >
= < χu(u, v) ∧ χv(u, v)∥χu(u, v) ∧ χv(u, v)∥
, χuv(u, v) >
= (χu(u, v), χv(u, v), χuv(u, v))
∥χu(u, v) ∧ χv(u, v)∥
χuv(u, v) = (−f ′(u)sen v, f ′(u)cos v, 0)
(χu(u, v), χv(u, v), χuv(u, v)) =
f ′(u)cos v −f (u)sen v −f ′(u)sen vf ′(u)sen v f (u)cos v f ′(u)cos v
g′(u) 0 0
= −f ′(u)f (u)g′(u)sen v cos v + f ′(u)f (u)g′(u)sen v cos v= 0
f (u, v) = 0.
g(u, v) = < N (u, v), χvv(u, v) >
= < χu(u, v) ∧ χv(u, v)∥χu(u, v) ∧ χv(u, v)∥ , χvv(u, v) >
= (χu(u, v), χv(u, v), χvv(u, v))
∥χu(u, v) ∧ χv(u, v)∥
χvv(u, v) = (−f (u)cos v, −f (u)sen v, 0)
(χu(u, v), χv(u, v), χvv(u, v)) =
f ′(u)cos v −f (u)sen v −f (u)cos vf ′(u)sen v f (u)cos v −f (u)sen v
g′(u) 0 0
= f (u)2g′(u)sen2 v + f (u)2g′(u)cos2 v
= f (u)2g′(u)
∥χu(u, v) ∧ χv(u, v)∥ = f (u)
g(u, v) = f 2
(u)g′(u)f (u)
= f (u)g′(u)
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K (u, v) = e(u, v)g(u, v) − f 2(u, v)E (u, v)G(u, v) − F 2(u, v)
=
(f ′(u)g′′(u)
−f ′′(u)g′(u))(f (u)g′(u))
−02
1f (u)2 − 02=
(f ′(u)g′′(u) − f ′′(u)g′(u))(f (u)g′(u))f (u)2
= (f ′(u)g′′(u) − f ′′(u)g′(u))g′(u)
f (u)
f ′(u)2 + g′(u)2 = 1
2f ′(u)f ′′(u) + 2g′(u)g′′(u) = 0
f ′(u)f ′′(u) + g ′(u)g′′(u) = 0
g′(u)g′′(u) = −f ′(u)f ′′(u)
(f ′(u)g′′(u) − f ′′(u)g′(u))g′(u) = f ′(u)g′(u)g′′(u) − f ′′(u)g′(u)2= −f ′(u)f ′(u)f ′′(u) − f ′′(u)g′(u)2=
−f ′(u)2f ′′(u)
−f ′′(u)g′(u)2
= −f ′′(u)(f ′(u)2 + g′(u)2)= −f ′′(u)
K (u, v) = −f ′′(u)
f (u)
χ(u, v) = (f (u)cos v, f (u)sen v, g(u))
f (u) > 0
f ′(u)2 + g′(u)2 = 1
(3.22)
K (u, v) = −f ′′(u)
f (u)
K (u, v) = 0
∀ (u, v) f ′′(u) = 0
f ′(u) = a
f (u) = au + b
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f ′(u)2 + g ′(u)2 = 1
g′(u) = ±√ 1 − a2 |a| ≤ 1
g(u) = ±√ 1 − a2u + c
c
χ(u, v) = ((au + b)cos v, (au + b)sen v, ±√
1 − a2u + c)
x = x1
y = y1
z = z 1 + c
χ(u, v) = ((au + b)cos v, (au + b)sen v, ±√ 1 − a2u)
z 1 = −√
1 − a2u
x1 = x2
y1 = y2z 1 = −z 2
χ(u, v) = ((au + b)cos v, (au + b)sen v,√
1 − a2u)
a = 0
χ(u, v) = (b cos v, b sen v, u)
b > 0
f (u) = b > 0
x22b2
+ y22b2
= 1
|a| = 1 χ(u, v) = ((±u+b)cos v, (±u+b)sen v, 0)
Ox2y2
z 2 = 0
0
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X = x3
Y = y3
Z = z 3−
b√
1 − a2
a
x23a2
1 − a2+
y23a2
1 − a2= z 23
K (u, v) = 1
(u, v)
f ′′
(u) + f (u) = 0
λ2 + 1 = 0
λ = ±i
f ′′(u) + f (u) = 0
eiu = cos u + i sen u
f ′′(u) + f (u) = 0
f (u) = A cos u + B sen u
A2 + B 2 = a2 > 0
(Aa
)2 + ( Ba
)2 = 1
Aa
= cos b
Ba
= − sen b
f (u) = a cos b cos u − a sen b sen u= a(cos u cos b − sen u sen b)= a cos(u + b), a ̸= 0.
f ′(u)2 + g′(u)2 = 1
g′(u)2 = 1 − (−a sen(u + b))2= 1 − a2 sen2(u + b)
g′(u) = ±√
1 − a2 sen2(u + b)
g(u) = ± √ 1 − a2 sen2(u + b)du + c
χ(u, v) = (a cos(u+b)cos v, a cos(u+b)sen v, ± √
1 − a2 sen2(u + b)du+c)
a ̸= 0
ũ = u + b v = ṽ
χ(ũ, ṽ) = (a cos ũ cos ṽ, a cos ũ sen ṽ, ± √
1 − a2 sen2 ũdũ + c), a ̸= 0
x = x1y = y1
z = z 1 + c
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χ(ũ, ṽ) = (a cos ũ cos ṽ, a cos ũ sen ṽ, ± √
1 − a2 sen2 ũdũ), a ̸= 0
z 1 = − √ 1 − a2 sen2 ũdũ x1 = x2
y1 = y2
z 1 = −z 2
χ(ũ, ṽ) = (a cos ũ cos ṽ, a cos ũ sen ṽ, √
1 − a2 sen2 ũdũ), a ̸= 0
√ 1 − a2 sen2 ũdũ
a = ±1 a ̸= 0
a = −1
a = 1
x2 = −x3y2 = −y3z 2 = z 3
a = 1
χ(ũ, ṽ) = (cos ũ cos ṽ, cos ũ sen ṽ,
√ 1 − sen2 ũdũ).
f (ũ) = cos ũ > 0 √
1 − sen2 ũdũ = √
cos2 ũdũ
=
cos ũdũ = sen ũ
χ(ũ, ṽ) = (cos ũ cos ṽ, cos ũ sen ṽ, sen ũ)
x23 + y
23 + z
23 = 1
K (u, v) = −1
f ′′(u) − f (u) = 0
λ2 − 1 = 0 ±1
f ′′(u) − f (u) = 0
eu
e−u
f (u) = aeu + be−u
f ′(u)2 + g′(u)2 = 1
g′(u) =
±√ 1 − (aeu − be−u)2 g(u) = ±
√ 1 − (aeu − be−u)2du + c
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c
g(u)
√ 1 − (aeu − be−u)2du
a = 1
b = 0
f (u) = eu
g(u) = ±
√ 1 − e2udu + c u ≤ 0
x = x1
y = y1
z = z 1
χ(u, v) = (eu cos v, eusinv, ± √
1 − e2udu) u ≤ 0.
z 1 = −
√ 1 − e2udu
x1 = x2
y1 = y2
z 1 = −
z 2
χ(u, v) = (eu cos v, eu sen v,
√ 1 − e2udu), u ≤ 0
w = eu
√ 1 − e2udu =
√ 1 − w2
w dw
= 1 − w2w√ 1 − w2
dw
=
(
1
w − w) 1√
1 − w2 dw
=
1
w√
1 − w2 dw −
w√ 1 − w2 dw
A =
1
w√
1 − w2 dw B =
w√ 1 − w2 dw
x = w−1
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A =
w−1√
1 − w2 dw
= − x√ 1
−x−2
dx
x2
= −x−1√
1 − x−2 dx
=
− x
−1 1 − 1
x2
dx
=
− x
−1 x2 − 1
x2
dx
= − x−1√ x2−1x
dx
= −
1√ x2 − 1dx
= − cos h−1x= − cos h−1w−1= − cos h−1(e−u)
y = 1 − w2
B =
w√
1 − w2 dw
= − 12√ y dy
= −12
y−
12 dy
= −12
y12
1
2= −√ y= −√ 1 − w2= −√ 1 − e2u
.
√ 1 − e2udu = − cos h−1(e−u) +
√ 1 − e2u + d
χ(u, v) = (eu cos v, eu sen v,√
1 − e2u − cos h−1(e−u) + d), u ≤ 0,
d
x2 = x3y2 = y3
z 2 = z 3 + d
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χ(u, v) = (eu cos v, eu sen v,√
1 − e2u − cos h−1(e−u)), u ≤ 0,
α(u) = (eu,
√ 1 − e2u − cos h−1(e−u)), u ≤ 0
α(u)
Oxz
z =√
1 − x2 − cos h−1( 1x
)
0 < x ≤ 1
H
H
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χ
N
λ
χλ
χ
χλ = χ + λN
χλ
χ
λ
χ
N
k1 k2
χ : U → R3
C |k1| |k2|
≤ C
λ
|λ| < 1C
χλ
χ
χλ
N λ
χλ
χλ(u, v)
N
χ
χ(u, v)
(u, v) ∈ U
χλ
k1(1−λk1)
k2(1−λk2)
χ
k1 k2
χλ
K 1−2λH +λ2K
H −λK 1−2λH +λ2K
χλ
χλ
χλ : U → χλ(U )
χλu ∧ χλv ̸= 0
χλ
χ
N
(χλ)−1 : χλ(U )
→ U
(χλ)−1(χλ( p)) = χ−1(χλ( p)
−λN ( p))
p ∈ U (χλ)−1
χλ ◦ (χλ)−1(χλ( p)) = χλ(χ−1(χλ( p) − λN ( p))) = χλ(χ−1(χ( p))) = χλ( p)
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(χλ)−1 ◦ χλ( p) = (χλ)−1(χλ( p)) = χ−1(χλ( p) − λN ( p)) = χ−1(χ( p)) = p
χλ
χλu ∧ χλv ̸= 0
χλu = χu + λN u
χλv = χv + λN v
N u = a11χu + a21χv
N v = a12χu + a22χv
χλu = χu + λ(a11χu + a21χv) = (1 + λa11)χu + λa21χv
χλv = χv + λ(a12χu + a22χv) = λa12χu + (1 + λa22)χv
χλu ∧ χλv = [(1 + λa11)(1 + λa22) − λ2a12a21]χu ∧ χv= [(a11a22 − a12a21)λ2 + (a11 + a22)λ + 1]χu ∧ χv= [(k1k2)λ
2 − (k1 + k2)λ + 1]χu ∧ χv= [(1
−λk1)(1
−λk2)]χu
∧χv
̸= 0
χu ∧ χv ̸= 0 |ki| ≤ C, i = 1, 2 |λ| < 1C
λki ≤ |λki| < 1
1 − λki > 0, i = 1, 2
N λ = χλu ∧ χλv∥χλu ∧ χλv∥
= [(1 − λk1)(1 − λk2)]χu ∧ χv|(1 − λk1)(1 − λk2)| ∥χλu ∧ χλv∥
= (1
−λk
1)(1
−λk
2)
(1 − λk1)(1 − λk2) . χu
∧χv
∥χu ∧ χv∥=
χu ∧ χv∥χu ∧ χv∥
= N
χλ = χ + λN
N λ = N
N λu = N u = a11χu + a21χv
N λv = N v = a12χu + a22χv
χλu = (1 + λa11)χu + λa21χv
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χλv = λa12χu + (1 + λa22)χv
1 + λa11 λa21
λa12 1 + λa22
χu
χv
=
χλu
χλv
χu
χv
=
1
det
1 + λa11 λa21
λa12 1 + λa22
1 + λa22 −λa21−λa12 1 + λa11
χλu
χλv
= 1
(1 − λk1)(1 − λk2) 1 + λa22 −λa21−λa12 1 + λa11
χλ
uχλv
= 1
(1 − λk1)(1 − λk2)
1 0
0 1
+ λ
a22 −a21−a12 a11
χλu
χλv
N λ = N
N λu
N λv =
a11 a21
a12
a22
χu
χv
a11 a12
a21 a22
dN p −k1 −k2
N λuN λv
= a11 a21
a12 a22 1
(1 − λk1)(1 − λk2) 1 00 1 + λ a22 −a21−a12 a11 χλu
χλv
= 1
(1 − λk1)(1 − λk2)
a11 a21
a12 a22
1 0
0 1
+ λ
a22 −a21−a12 a11
χλu
χλv
1
(1 − λk1)(1 − λk2)
a11 a21
a12 a22
1 0
0 1
+ λ
a22 −a21−a12 a11
t
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dN λ p −kλ1 −kλ2
1(1 − λk1)(1 − λk2)
a11 a21
a12 a22 1 0
0 1 + λ a22 −a21−a12 a11
t
= 1
(1 − λk1)(1 − λk2)
1 0
0 1
+ λ
a22 −a12−a21 a11
a11 a12
a21 a22
v
a11 a12
a21 a22
(−k1) (−k2) v
1
(1 − λk1)(1 − λk2)
1 0
0 1
+ λ
a22 −a12−a21 a11
a11 a12
a21 a22
−k11−λk1
−k21−λk2
−k1 −k2
a11 a12
a21 a22
v = −k1v
1
(1 − λk1)(1 − λk2)
1 0
0 1
+ λ
a22 −a12−a21 a11
a11 a12
a21 a22
v
= −k1(1 − λk1)(1 − λk2)
1 00 1
v + λ
a22 −a12−a21 a11
v
v =
a
b
a11 a12
a21 a22
a
b
= −k1
a
b
a11a + a12b = −k1a
a21a + a22b = −k1b
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a22 −a12−a21 a11
a
b
=
a22a − a12b−a21a + a11b
(3.25);(3.26)=
a22a + k1a + a11ak1b + a22b + a11b
=
(k1 + a22 + a11)a
(k1 + a22 + a11)b
=
(−k1 − k2 + k1)a
(k1 − k1 − k2)b
=
−k2a−k2b
= −k2 ab = −k2v
1
(1 − λk1)(1 − λk2)
1 0
0 1
+ λ
a22 −a12−a21 a11
a11 a12
a21 a22
v
= −k1
(1 − λk1)(1 − λk2) [v + λ(−k2v)] = −k1
(1 − λk1)(1 − λk2) (1−λk2)v = −k1(1 − λk1)v
χλ
kλ1 =
k1
(1 − λk1) kλ2 =
k2
(1 − λk2)
K λ = k1
(1 − λk1)k2
(1 − λk2)=
k1k2
1 − λk2 − λk1 + λ2k1k2=
k1k2
1 − λ(k1 + k2) + λ2k1k2=
K
1 − 2λH + λ2K
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H λ =
k1
(1 − λk1) + k2
(1 − λk2)2
=
k1(1
−λk2) + k2(1
−λk1)
(1 − λk1)(1 − λk2)2
= k1 − λk1k2 + k2 − λk1k2
2(1 − λk1)(1 − λk2)=
k1 + k2 − 2λk1k22(1 − λ(k2 + k1) + λ2k1k2)
= 2H − 2λK
2(1 − 2λH + λ2K )=
H − λK 1 − 2λH + λ2K
χ
H ̸= 0 λ = 1
2H χλ
4H 2 > 0
χ
K > 0
K ̸= H 2
λ = ± 1√ K
χλ
∓12
√ K ̸= 0
λ = 12H
K λ = K
1 − 2λH + λ2K =
K
1 − 2 12H
H + ( 12H
)2K
= K
1 − 1 + K 4H 2
= K
K 4H 2
= 4H 2
.
λ =
1
√ K
H λ = H − λK
1 − 2λH + λ2K =
H − 1√ K
K
1 − 2 1√ K
H + ( 1√ K
)2K
=
H √ K −K √ K
1 − 2H √ K
+ K K
=
H √ K −K √ K
2
− 2H √
K
=
H √ K −K √ K
2√ K −2H √ K
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= H
√ K − K
2√
K − 2H = −1
2
(K − H √ K )(√
K − H )= −
1
2
K
−H √ K
(√ K −H )√ K√ K
= −12
K − H √ K K −H
√ K √
K
= −12
√ K
.
λ = − 1√ K
S
P
S
K
> 0
X
R3
X ⊂ R3
f : R3 → R
P
Q
X
f (Q) ≤ f (R) ≤ f (P )
R
X
f
X
P
Q
6.4
f : R3 → R
f (v) = ∥v∥2
f
S
P
S
f
P
S
∥
∥
S
P
S
P
∥
∥
1∥ ∥2
χ(u, v) =
∥ ∥ cos u∥ ∥ cos v, ∥ ∥ cos u∥ ∥ sen v, ∥ ∥ sen u∥ ∥
cos u∥
∥ > 0
γ (t)
S
P
t = 0
f (γ (t))
t = 0
d
dtf (γ (t)) |t=0 = 0 d
2
dt2f (γ (t)) |t=0 ≤ 0.
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f ′(γ (t))γ ′(t) |t=0 = 0 f ′′(γ (t))(γ ′(t))2 + f ′(γ (t))γ ′′(t) |t=0 ≤ 0
f (v) = ∥v∥2 =< v, v >
f ′(v) = 2 < v′, v >
f ′′(v) = 2(< v′′, v > + < v′, v′ >).
0 = f ′(γ (0))γ ′(0) = 2 < γ ′(0), γ (0) > γ ′(0)
0 ≥ f ′′(γ (0))(γ ′(0))2 + f ′(γ (0))γ ′′(0) =2(< γ ′′(0), γ (0) > + < γ ′(0), γ ′(0) >)(γ ′(0))2 + 2 < γ ′(0), γ (0) > γ ′′(0).
γ (t)
γ ′(t) ̸= 0 t < γ ′(t), γ ′(t) >= 1 γ ′(0) ̸ = 0 < γ ′(0), γ ′(0) >= 1
< γ ′(0), γ (0) >= 0 (< γ ′′(0), γ (0) > +1)(γ ′(0))2 ≤ 0
< γ ′(0), γ (0) >= 0
< γ ′′(0), γ (0) > +1 ≤ 0
= γ (0)
S
P
S
P
Y S P N
N = ± ∥
∥
kn(P ) =< γ ′′(0), N >
γ
P
kn(P ) ≤ −1∥ ∥ N =
∥ ∥
kn(P )
≥
1
∥ ∥
N = −
∥ ∥
N = ∥
∥ k1 k2 P
k2 ≤ kn(P ) ≤ k1 ≤ −1∥ ∥
N = − ∥ ∥ k1 k2 P 1∥
∥ ≤ k1 ≤ kn(P ) ≤ k2 Y P ≤ −1∥ ∥ ≥ 1∥ ∥
K = k1k2 ≥ 1∥
∥2 > 0 P
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χ : U ⊂ R2 → R3
D ⊂ U h : D → R D D
∂D
χ(D)
h
ϕ : D×] − ϵ, ϵ[→ R3
ϕ(u , v , t) = χ(u, v) + th(u, v)N (u, v)
(u, v) ∈ D
t ∈] − ϵ, ϵ[.
t ∈] − ϵ, ϵ[
χt : D → R3
χt(u, v) = ϕ(u , v , t)
∂χt
∂u
= χu + thN u + thuN
∂χt
∂v = χv + thN v + thvN.
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E t =< χtu, χtu >,
F t =< χtu, χtv >
Gt =< χtv, χtv >
E t = < χtu, χtu >
= < χu + thN u + thuN, χu + thN u + thuN >
= < χu, χu > + < χu,thN u > + < thN u, χu > + < thN u,thN u >
+ < thuN,thuN >
= E + 2th < χu, N u > +t2h2 < N u, N u > +t
2h2u
F t = < χtu, χtv >
= < χu + thN u + thuN, χv + thN v + thvN >
= < χu, χv > + < χu,thN v > + < thN u, χv > + < thN u,thN v >
+ < thuN,thvN >
= F + th(< χu, N v > + < N u, χv >) + t2h2 < N u, N v > +t
2huhv
Gt = < χtv, χtv >
= < χv + thN v + thvN, χv + thN v + thvN >
= < χv, χv > + < χv,thN v > + < thN v, χv > + < thN v,thN v >
+ < thvN,thvN >
= G + 2th < χv, N v > +t2h2 < N v, N v > +t
2h2v
−e = < χu, N u >−2f = < χu, N v > + < χv, N u >−g = < χv, N v >
H =
Eg
−2F f + Ge
2(EG − F 2) .
E t = E − 2the + t2h2 < N u, N u > +t2h2uF t = F − 2thf + t2h2 < N u, N v > +t2huhvGt = G − 2thg + t2h2 < N v, N v > +t2h2v
E tGt − (F t)2 = EG − F 2 − 2th(Eg − 2F f + Ge) + R= (EG − F 2) − 2th[2H (EG − F 2)] + R
= (EG
−F 2
)(1 − 4thH
) +R
limt→0
R
t
= 0
R
t2
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χtu ∧ χtv2 = E tGt − (F t)2 = (EG − F 2)(1 − 4thH ) + R
limt→0R
t
= 0.
ϵ > 0
χt
limt→0
χtu ∧ χtv2 = limt→0
(EG − F 2)(1 − 4thH ) + R= EG − F 2 + lim
t→0R
= EG − F 2 + limt→0
t(R
t )
= EG − F 2 > 0
ϵ > 0
χtu∧
χtv ̸= 0
A(t)
χt( D̄)
A(t) =∫ ∫ ̄
D
√ E tGt − (F t)2dudv
=∫ ∫ ̄
D
√ (EG − F 2)(1 − 4thH ) + Rdudv
=∫ ∫ ̄
D
(EG − F 2)(1 − 4thH ) + R
EG−F 2 (EG − F 2)dudv=
∫ ∫ ̄D
√ (EG − F 2)(1 − 4thH + R̄)dudv
R̄ = R
(EG−F 2)
A(t) =
D̄
√ 1 − 4thH + R̄√ EG − F 2dudv
ϵ
A
t = 0
A′(0) = limt→0
A(t) − A(0)t
= limt→0
D̄
√ 1 − 4thH + R̄ −
√ 1 + R̄
√ EG − F 2
t dudv
=∫ ∫ ̄
D
limt→0 (
√ 1 − 4
thH +
R̄−
√ 1 +
R̄)t√ EG − F 2dudv
limt→0
√ 1 − 4thH + R̄ −
√ 1 + R̄
t =
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= limt→0
√ 1 − 4thH + R̄ −
√ 1 + R̄
√ 1 − 4thH + R̄ +
√ 1 + R̄
t√
1 − 4thH + R̄ +√
1 + R̄
= limt→
0
1 − 4thH + R̄ − 1 − R̄
t√ 1 − 4thH + R̄ + √ 1 + R̄
= limt→0
−4hH √ 1 + R̄ +
√ 1 + R̄
= limt→0
−4hH 2√
1 + R̄
= −4hH
2= −2hH
limt→0
R̄ = limt→0
R
(EG
−F 2)
= 0
limt→0
R = 0
A′(0) =
D̄
−2hH √
EG − F 2dudv
γ R
3
t ∈]−ϵ, ϵ[
χt : D → R3 D = int(π) π
U
γ = χt ◦ π
A(t) =
D̄
√ 1 − 4thH + R̄
√ EG − F 2dudv
χ = χo
γ
A
t = 0
A′(0) = 0
0 =
∫ ∫ ̄D
2hH √
EG − F 2dudv h : D̄ → R
h : D̄ → R h(q ) = H (q ) q ∈ D̄ hH = H 2 0 =
∫ ∫ ̄D
2H 2√
EG − F 2dudv
√ EG − F 2 > 0
2H 2
√ EG − F 2 ≥ 0
H ≡ 0
γ
R3
S
γ
S
χ
S
A′(0) = 0
H ≡ 0
S
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x = 1a
cosh az
Oxz
Oz
a
a = 1
χ(u, v) = (cosh u cos v, cosh u sen v, u), 0 < v
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F = < χu, χv >
= − senh u cosh u sen v cos v + senh u cosh u sen v cos v= 0
G =
< χv
, χv
>
= cosh2 u sen2 v + cosh2 u cos2 v + 0
= cosh2 u(sen2v + cos2v)
= cosh2 u
e = < N,χuu >
= − u cosh u cos2 v − u cosh u sen2 v= −
u cosh u(sen2 v + cos2 v)
= −
u cosh u
= −
1
coshu
cosh u
= −1f = < N,χuv >
=
u senh u sen v cos v −
u senh u sen v cos v
= 0
g = < N,χvv >
=
u cosh u cos2 v +
u cosh u sen2 v
=
u cosh u(sen2 v + cos2 v)
=
u cosh u= 1
cosh u cosh u
= 1
H = eG−2fF +gE 2(EG−F 2)
= (−1) cosh2 u+cosh2 u
2(cosh2 u cosh2 u−02)= 0
2cosh4 u
= 0
S
S
z
Oxz
S
χ(u, v) = (f (u)cos v, f (u)sen v, g(u)),
α(u) = (f (u), 0, g(u))
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f (u) > 0
E = 1
F = 0G = f 2(u)
e = f ′(u)g′′(u) − f ′′(u)g′(u)f = 0
g = f (u)g′(u)
H = eG−2fF +gE 2(EG−F 2)
H = [f ′(u)g′′(u) − f ′′(u)g′(u)]f 2(u) + f (u)g′(u)
2f 2(u)
= 12 [f ′(u)g′′(u) − f ′′(u)g′(u) + g′(u)f (u) ]
u
u = u0 g
′(u0) ̸= 0
g′(u) ̸= 0
u
u0
]α, β [
u ∈]α, β [. α
(f ′(u))2 + (g′(u))2 = 1.
f ′(u)g′′(u) − f ′′(u)g′(u) = −f ′′(u)
g′(u) .
H = 1
2
g′(u)f (u)
− f ′′(u)
g′(u)
(g′(u))2 = 1 − (f ′(u))2
S
H = 0
(g′(u))2
−f (u)f ′′(u) = 0
f (u)f ′′(u) = 1 − (f ′(u))2
f f ′′ = 1 − (f ′)2
h = df du
f ′′ = dh
du = dh
df · df
du = hdh
df
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hf dh
df = 1 − h2
g′(u) ̸= 0 h2
+ (g′)2
= 1
1 − h2
= (g′)2
> 0
h
1 − h2 dh =
df
f + c
c = lna
a > 0
x = 1 − h2 > 0
dx = −2hdh
h1 − h2
dh = dx−2x = −1
2 lnx =
−1
2 ln(1
−h2) = ln
1
√ 1 − h2.
ln 1√ 1 − h2 = lnf + lna
ln 1√ 1 − h2 = lnaf,
1√ 1 − h
2= af
1 − h2 = ( 1af
)2
h2 = 1 − ( 1af
)2
= a2f 2−1a2f 2
,
|h| =√
a2f 2 − 1af
, a > 0.
h > 0
f (u)
−f (u)
h −h > 0
h = √ a2f 2−1af
h = df
du
df
du =
√ a2f 2 − 1
af ,
af
√ a2f 2 − 1 df = du.
af √
a2f 2 − 1 df =
du.
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x = a2f 2 − 1 dx = 2a2f df dx
2a = af df
∫ af √ a2f 2−1
df =∫
dx2a√ x
= 12a ∫ x− 12 dx
= 2 12a
x12
= 1a
x12
= 1a
√ a2f 2 − 1
√ a2f 2 − 1
a = u + b,
b
a2f 2 − 1 = a2(u + b)2
a2f 2 = a2(u + b)2 + 1
f 2 = 1a2
[a2(u + b)2 + 1]
f = |f | =√
a2(u + b)2 + 1
a ,
b
u → u − b b = 0
f = √ a2
u2
+ 1a
.
g
(g′)2 = 1 − (f ′)2 = 1 − h2 = 1a2f 2
dg
du = ± 1√
a2u2 + 1
g = ±
1√ a2u2 + 1
du + c,
c
∫ 1√ a2u2+1
du.
u = senhxa
du = 1a
cosh xdx
∫ 1√ a2u2+1
du =∫
1√ (senhx)2+1
1a
cosh xdx
= 1a
∫ 1√
(coshx)2 cosh xdx
= 1a
∫ dx
= xa
.
senh x = au
x = arcsenh(au)
1√ a2u2 + 1
du = 1
a arcsenh(au).
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g = ±arcsenh(au) + caa
.
a(g − c) = ± arcsenh(au),
arcsenh(au) = ∓a(g − c),
au = senh[∓a(g − c)]= ∓ senh[a(g − c)] .
f = √ (senh[a(g − c)])2 + 1
a= 1a
√ (cosh[a(g − c)])2
= 1a
cosh[a(g − c)].
S
α(u) = (1
a cosh[a(g(u) − c)], 0, g(u))
Oxz
x = 1
a cosh[a(z − c)].
z c = 0
x = 1
a cosh(az )
S
u ∈]α, β [
g′(u) ̸= 0.
β α
S
u ∈ (α, ∞)
α < ∞
u ≤
α
g′(α) = 0
]α, β [
u0
g′(u) ̸= 0.
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β
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∥χu ∧ χv∥ =√
α2 + u2ω2
N = χu ∧ χv∥χu ∧ χv∥ =
(α sen ωv, −α cos ωv,uω)√ α2 + u2ω2
χuu = (0, 0, 0)
χuv = (−ω sen ωv,w cos ωv, 0)χvv = (−uω2 cos ωv, −uω2 sen ωv, 0)
E = < χu, χu >
= cos2 ωv + sen2 ωv
= 1
F = < χu, χv >
= −uω sen ωv cos ωv + uω sen ωv cos ωv= 0
G = < χv, χv >
= u2ω2 sen2 ωv + u2ω2 cos2 ωv + α2
= u2ω2 + α2
e = < N,χuu >
= < N, (0, 0, 0) >
= 0
f = < N,χuv >
= −αω sen2 ωv − αω cos2 ωv√
α2 + u2ω2
= −αω√
α2 + u2ω2
g = < N,χvv >
= −αuω2 sen ωv cos ωv + αuω2 sen ωv cos ωv√
α2 + u2ω2
= 0
H = eG−2fF +gE 2(EG−F 2)
= 0G−2f 0+0E 2(EG−F 2)
= 0
C
R
3
P
C
Q
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γ
C
γ (u) = Q
δ (u)
γ (u)
P
χ(u, v) = γ (u) + vδ (u)
χu = γ ′(u) + vδ ′(u)
χv = δ (u).
χ
γ ′(u) + vδ ′(u)
δ (u)
γ ′
δ
v
C
χ(u, v) = γ (u) + vδ (u)
γ
δ (u)
γ (u).
∥δ (u)∥ = 1 u
δ ′(u) ̸= 0
u
δ ′(u) = 0
u
δ
γ
∥δ (u)∥ = 1 < δ (u), δ (u) >= 1 < δ ′(u), δ