Tải bản đầy đủ

Các mặt không định hướng được

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❈⑩❈ ▼➄❚ ❑❍➷◆● ✣➚◆❍ ❍×❰◆● ✣×Ñ❈

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽


❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤


❈⑩❈ ▼➄❚ ❑❍➷◆● ✣➚◆❍ ❍×❰◆● ✣×Ñ❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❍➻♥❤ ❤å❝
▼➣ sè✿ ❄❄❄❄❄❄❄

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿

P●❙✳❚❙✳ ◆❣✉②➵♥ ❚❤↕❝ ❉ô♥❣

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✽




r tớ ự t õ tổ ữủ sỹ q
t ở ừ t ổ tr tờ ồ õ r
trữớ ồ ữ ở õ ũ ợ sỹ ộ trủ ú ù ừ
s
ổ t ỡ t ổ tr trữớ ồ ữ
ở t ổ t t ú ù tổ tr ố ồ ứ q
t tổ t õ
t tổ ữủ tọ ỏ t ỡ s s t P
ụ t t ữợ ú ù tổ tr sốt q tr tỹ
õ
ỏ tr ở tớ ỳ tr tr õ
ổ tr ọ ỳ t sõt tổ rt ữủ sỹ ú ù
õ ỵ ừ t ổ s õ ừ tổ õ t t
ỡ ỳ
ổ t ỡ
ở t





▲❮■ ❈❆▼ ✣❖❆◆

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔② ❝õ❛ tæ✐✱ ✤÷ñ❝ ❤➻♥❤ t❤➔♥❤ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛②
❣✐→♦ P●❙✳❚❙✳ ◆❣✉②➵♥ ❚❤↕❝ ❉ô♥❣ ❝ò♥❣ ✈î✐ ✤â ❧➔ sü ❝è ❣➢♥❣ ❝õ❛ ❜↔♥ t❤➙♥✳
❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ tæ✐ ✤➣ t❤❛♠ ❦❤↔♦ ✈➔ t❤ø❛ ❦➳ ♥❤ú♥❣ t❤➔♥❤ q✉↔ ♥❣❤✐➯♥
❝ù✉ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝✱ ✤➦❝ ❜✐➺t ❧➔ ❝õ❛ ❆✳●r❛② tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶❪✱ ✈î✐ sü tr➙♥ trå♥❣
✈➔ ❧á♥❣ ❜✐➳t ì♥✳
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉
❝õ❛ r✐➯♥❣ ❜↔♥ t❤➙♥ ❞ü❛ tr➯♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✱ ❦❤æ♥❣ ❝â sü trò♥❣ ❧➦♣ ✈î✐ ❦➳t
q✉↔ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❦❤→❝✳
❍➔ ◆ë✐✱ ✶✵ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤


▼ö❝ ❧ö❝
▲í✐ ♠ð ✤➛✉



✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à



✶✳✶

▼↔♥❤ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

▼↔♥❤ ❤➻♥❤ ❤å❝ ✈➔ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣

E3

✳ ✳ ✳ ✳ ✳ ✳

✷ ❚➼♥❤ ✤à♥❤ ❤÷î♥❣ ❝õ❛ ♠➦t




✶✶

✷✳✶

▼➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✷

❚r÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝

✳ ✳

✶✺

✷✳✸

❈→❝ ♠↔♥❤ ❝â ❝ò♥❣ ✤à♥❤ ❤÷î♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✸ ❈→❝ ♠➦t ❦❤æ♥❣ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝
✸✳✶

❈→❝ ♠➦t ♥❤➟♥ ✤÷ñ❝ ❜➡♥❣ ♣❤➨♣ ✤ç♥❣ ♥❤➜t

✸✳✷

❉↔✐ ▼♦❜✐✉s

✷✶
✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✸✳✸

❈❤❛✐ ❑❧❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✸✳✹

❙ü ❤✐➺♥ t❤ü❝ ❤â❛ ❝õ❛ ♠➦t ♣❤➥♥❣ ①↕ ↔♥❤ t❤ü❝

✳ ✳ ✳ ✳ ✳

✸✵

✸✳✺

▼➦t ①♦➢♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✵

✐✐




õ tốt ồ


ỵ ồ t
ồ ổ ồ t ỗ tứ tỹ t õ ự ử
rở r ũ ợ tớ tự ồ ổ ứ ữủ
ờ ợ tr ừ ởt tr ỳ ổ ồ õ
trỏ q trồ sỹ t tr ừ ồ õ ồ

ồ ổ ồ ữủ tr
ỳ tự ỡ ỵ tt ữớ ỵ tt t tr

En (n = 2, 3)
tr

E3

ố ữủ t s t t

ồ t

t ổ ữợ

ữủ õ tốt
ử ự
ữợ q ợ ự ồ t s
ỡ ồ t t ổ ữợ ữủ

ử ự
ự t ữợ ừ t t ổ
ữợ ữủ

Pữỡ ự
ồ t t tờ ủ

trú õ
ở õ ỗ ữỡ
ữỡ

tự



◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✧❚➼♥❤ ✤à♥❤ ❤÷î♥❣ ❝õ❛ ♠➦t✧
❈❤÷ì♥❣ ✸ ✧❈→❝ ♠➦t ❦❤æ♥❣ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✧
❈❤÷ì♥❣ ✷




ữỡ
tự
sốt tr t ở õ ú t sỷ ử ỵ
ổ tự ổ tỡ

R3

E3

ợ t

ổ ữợ t r ữỡ t s tố ởt số
ỡ ỵ tt t tr ổ

E3

t ú

t ợ t t số s õ tr ồ
t tr

E3

t ú

t t q

t số
r ử ú t s ỡ q
t số ỗ t số ữớ tồ
ở q q t t
tr

E3

rữợ t t õ t số



tử r tứ ởt t tr R2



◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❦❤æ♥❣ ❣✐❛♥ E3
r : U −→ E3
(u, v) → r(u, v)

✤÷ñ❝ ❣å✐ ❧➔ ♠ët ♠↔♥❤ t❤❛♠ sè tr♦♥❣ E3✳ ❚ò② t❤❡♦ ✤è✐ t÷ñ♥❣ ♥❣❤✐➯♥
❝ù✉✱ ♠➔ s❛✉ ♥➔② t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t ✈➲ ❝➜♣ ❦❤↔ ✈✐ ❝õ❛ r✳
❈❤♦ ♠ët ♠↔♥❤ t❤❛♠ sè

r : U → E3 ✱

t❛ ❝â ❦❤→✐ ♥✐➺♠ ✤÷í♥❣ tå❛ ✤ë

♥❤÷ s❛✉✳
❱î✐ ✤✐➸♠

(u0 , v0 ) ∈ U ✿

❈✉♥❣ t❤❛♠ sè
ð ✤➙②

u

v

tr♦♥❣

t❤❛② ✤ê✐ tr♦♥❣ ♠ët ❦❤♦↔♥❣

❈✉♥❣ t❤❛♠ sè
ð ✤➙②

u → r(u, v0 )

u → r(u0 , v)

❣å✐ ❧➔ ✤÷í♥❣ tå❛ ✤ë

I⊂R

tr♦♥❣

t❤❛② ✤ê✐ tr♦♥❣ ♠ët ❦❤♦↔♥❣

E3
E3

♥➔♦ ✤â✱

u0 ∈ I ✳

❣å✐ ❧➔ ✤÷í♥❣ tå❛ ✤ë

J ⊂R

♥➔♦ ✤â✱

v = v0 ✱

u = u0 ✱

u0 ∈ J ✳

❚✐➳♣ t❤❡♦✱ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② ✈➔ ✤✐➸♠ ❦➻ ❞à✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✣✐➸♠ (u0 , v0 ) ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉② ❝õ❛ ♠↔♥❤

t❤❛♠ sè r ♥➳✉ ❤➺ ❤❛✐ ✈❡❝tì {ru(u0, v0), rv (u0, v0)} ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳
◆❤ú♥❣ ✤✐➸♠ ❦❤æ♥❣ ❝❤➼♥❤ q✉② t❤➻ ✤÷ñ❝ ❣å✐ ❧➔ ❦➻ ❞à✳
▼↔♥❤ t❤❛♠ sè

r

✤÷ñ❝ ❣å✐ ❧➔ ❝❤➼♥❤ q✉② ♥➳✉ →♥❤ ①↕

r

❧➔ →♥❤ ①↕ ❦❤↔

✈✐ ❧✐➯♥ tö❝ ✈➔ ♠å✐ ✤✐➸♠ ❝õ❛ ♥â ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉②✳


− →
− →
❈❤♦ ❝ì sð trü❝ ❝❤✉➞♥ {→
i ; j ; k } ❝õ❛ E3 ✈➔ ✤✐➸♠ O ∈
E3 ✳ ❈❤♦ ♠↔♥❤ t❤❛♠ sè r : R2 −→ E3 ①→❝ ✤à♥❤ ❜ð✐

❱➼ ❞ö ✶✳✶✳✶✳

−−→


r(u, v) = O + R cos v e(u) + R sin v k
−−→




tr♦♥❣ ✤â R ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣✱ e(u) = cos u→
i + sin u j ✱ ❝â ↔♥❤ ❧➔




◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

♠➦t ❝➛✉ t➙♠ O✱ ❜→♥ ❦➼♥❤ R✱ ♠↔♥❤ t❤❛♠ sè r ❝â ❝→❝ ✤✐➸♠
u,

π
+ lπ ; l ∈ Z
2

❧➔ ✤✐➸♠ ❦➻ ❞à ❜ð✐ ✈➻ t↕✐ ❝→❝ ✤✐➸♠ ♥➔② t❛ ❝â ru = 0✳
❱î✐ u0 ∈ R ❝è ✤à♥❤✱ ✤÷í♥❣ tå❛ ✤ë u = u0 ❧➔ ✤÷í♥❣ ❦✐♥❤ t✉②➳♥✱ ✈î✐
v0 ∈ R ❝è ✤à♥❤✱ ✤÷í♥❣ tå❛ ✤ë v = v0 ❧➔ ✤÷í♥❣ ✈➽ t✉②➳♥ ❝õ❛ ♠➦t ❝➛✉✳
▼ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ tr✉♥❣ t➙♠ ❝õ❛ ❧þ t❤✉②➳t ♠➦t t❤❛♠ sè
❧➔ ❦❤→✐ ♥✐➺♠ t✐➳♣ ❞✐➺♥ ✈➔ ♣❤→♣ t✉②➳♥✳ ❚❛ ❝â ✤à♥❤ ♥❣❤➽❛ t✐➳♣ ❞✐➺♥ ✈➔
♣❤→♣ t✉②➳♥ ♥❤÷ s❛✉✳

❈❤♦ r ❧➔ ♠ët ♠↔♥❤ t❤❛♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ●✐↔
sû (u0, v0) ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉② ❝õ❛ ♠↔♥❤ t❤❛♠ sè r✳ ❑❤✐ ✤â ♠➦t ♣❤➥♥❣
tr♦♥❣ E3 ✤✐ q✉❛ r(u0, v0) ✈î✐ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝❤➾ ♣❤÷ì♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳




ru (u0 , v0 ), →
rv (u0 , v0 )

✤÷ñ❝ ❣å✐ ❧➔ ♠➦t ♣❤➥♥❣ t✐➳♣ ①ó❝ ❤❛② t✐➳♣ ❞✐➺♥ ❝õ❛ r t↕✐ r(u0, v0)✳
✣÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ✤✐➸♠ r(u0, v0) ✈✉æ♥❣ ❣â❝ ✈î✐ t✐➳♣ ❞✐➺♥ t↕✐ r(u0, v0)
✤÷ñ❝ ❣å✐ ❧➔ ♣❤→♣ t✉②➳♥ ❝õ❛ r t↕✐ r(u0, v0)✳
❈✉è✐ ❝ò♥❣✱ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✷ ♠↔♥❤ t❤❛♠ sè t÷ì♥❣ ✤÷ì♥❣✿
❚r♦♥❣

E3

❝❤♦ ❤❛✐ ♠↔♥❤ t❤❛♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝

r1 : U1 −→ E3
❚❛ ♥â✐ ❤❛✐ ♠↔♥❤

r1

✈➔

r2

✈➔

r2 : U2 −→ E3

❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❝â ♠ët ✈✐ ♣❤æ✐

λ : U1 −→ U2




◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
s❛♦ ❝❤♦

r1 = r2 ◦ λ✳

❑➼ ❤✐➺✉

r1 ∼ r2 ✳

✣â ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✱ ♠é✐ ❧î♣ t÷ì♥❣ ✤÷ì♥❣ ✤â ❣å✐ ❧➔
♠ët ♠↔♥❤ tr♦♥❣

E3

✈➔

r

❣å✐ ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ ♠↔♥❤✳

◆➳✉ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ♥â✐ tr➯♥✱ ✤á✐ ❤ä✐

λ

❧➔

♠ët ✈✐ ♣❤æ✐ ❜↔♦ t♦➔♥ ❤÷î♥❣ t❤➻ ❝â t❤➸ ♥â✐ ✤➳♥ ♠↔♥❤ ✤à♥❤ ❤÷î♥❣✳ ❑❤✐
✤â✱ tr♦♥❣

E3

✈➔ ♠↔♥❤ ❝❤➼♥❤ q✉② t❤➻ ✈❡❝tì ✤ì♥ ✈à

ru ∧ rv
ru ∧ rv
t↕✐ ✤✐➸♠ ù♥❣ ✈î✐

(u, v)

tr♦♥❣ ♠ët t❤❛♠ sè ❤â❛

r

❝õ❛ ♥â ❧➔ ❤♦➔♥ t♦➔♥

①→❝ ✤à♥❤ ✈➔ ♣❤÷ì♥❣ ❝õ❛ ♥â ❝❤➼♥❤ ❧➔ ♣❤÷ì♥❣ ❝õ❛ ♣❤→♣ t✉②➳♥ ❝õ❛ ♠↔♥❤
t↕✐ ✤✐➸♠ ✤â✳ Ð ✤➙②✱ ❦➼ ❤✐➺✉
✈➔

ru ∧ rv

❝❤➾ t➼❝❤ ❝â ❤÷î♥❣ ❣✐ú❛ ❤❛✐ ✈❡❝tì

ru

rv ✳

✶✳✷ ▼↔♥❤ ❤➻♥❤ ❤å❝ ✈➔ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ♥❤➢❝ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♠↔♥❤
❤➻♥❤ ❤å❝ ✈➔ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣

E3 ✳

❑❤→✐ ♥✐➺♠ trå♥❣ t➙♠ tr♦♥❣

♣❤➛♥ ♥➔② ❧➔ →♥❤ ①↕ ❦❤↔ ✈✐ ❣✐ú❛ ❤❛✐ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ ✈➔ →♥❤ ①↕ t✐➳♣
①ó❝✳ ❚❛ ❜➢t ✤➛✉ ✈î✐ ❦❤→✐ ♥✐➺♠ ♠↔♥❤ ❤➻♥❤ ❤å❝✳

❚➟♣ ❝♦♥ S ❝õ❛ E3 ✤÷ñ❝ ❣å✐ ❧➔ ♠↔♥❤ ❤➻♥❤ ❤å❝ tr♦♥❣
E3 ♥➳✉ S ❧➔ ↔♥❤ ❝õ❛ ♠↔♥❤ t❤❛♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ r : U −→ E3 ✱ s❛♦
❝❤♦ ✈î✐ ♠å✐ (u0, v0) ∈ U ✱ t❛ ❝â
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳

{ru (u0 , v0 ), rv (u0 , v0 )}

✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈➔ →♥❤ ①↕ r ❧➔ ♠ët ✤ç♥❣ ♣❤æ✐ tø U ❧➯♥ ↔♥❤ S = r(U )✳



◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
◆➳✉ ❙ ❧➔ ♠ët ♠↔♥❤ ❤➻♥❤ ❤å❝ t❤➻ →♥❤ ①↕
t❤❛♠ sè ❤â❛ ❝õ❛ ♠↔♥❤ ❤➻♥❤ ❤å❝

❱➼ ❞ö ✶✳✷✳✶✳

r

♥❤÷ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët

S✳

▲➜② tå❛ ✤ë ❛❢✐♥ (x1, x2, x3) ❝õ❛ E3 ✈➔ ①➨t →♥❤ ①↕
r : U −→ E3

①→❝ ✤à♥❤ ❜ð✐
(u, v) → r(u, v) = (u, v, x3 (u, v))

tr♦♥❣ ✤â U ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ R2 ✈➔ (u, v) → x3(u, v) ❧➔ ♠ët ❤➔♠
sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝❤♦ tr÷î❝ tr➯♥ U ✳
❘ã r➔♥❣ r ❧➔ ♠ët →♥❤ ①↕ ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ❉➵ t❤➜②
ru = (1, 0, (x3 )u )✱ rv = (1, 0, (x3 )v ).

❉♦ ✈➟②✱ {ru(u0, v0), rv (u0, v0)} ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈î✐ ♠å✐ (u0, v0) ∈
U ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ r✱ t❛ t❤➜② r ❧➔ ♠ët ✤ì♥ →♥❤✳ ❍ì♥ ♥ú❛✱ →♥❤ ①↕
♥❣÷ñ❝ r−1 : r(U ) −→ U ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ✈➻ ♥➳✉ ❤❛✐ ✤✐➸♠ (u1, v1, x3(u1, v1))
✈➔ (u2, v2, x3(u2, v2)) ❣➛♥ ♥❤❛✉ tr♦♥❣ E3 t❤➻ u1❀ u2 ❝ô♥❣ ♥❤÷ v1❀ v2 ♣❤↔✐
❣➛♥ ♥❤❛✉ tr♦♥❣ U ✳
❉♦ ✤â r(u, v) ❧➔ ♠ët ♠↔♥❤ ❤➻♥❤ ❤å❝ tr♦♥❣ E3✳
▲÷✉ þ r➡♥❣

r(u, v)

(u, v) → x3 (u, v)✳

❝❤➼♥❤ ❧➔ ✤ç t❤à ❝õ❛ ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝

❉♦ ✈➟②✱ ❝❤ó♥❣ t❛ ❝â ♥❤➟♥ ①➨t ✤ç t❤à ❝õ❛ ♠ët ❤➔♠

sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❧➔ ♠ët ♠↔♥❤ ❤➻♥❤ ❤å❝✳

S ❧➔ ♠ët ♠↔♥❤ ❤➻♥❤ ❤å❝ ✈î✐ t❤❛♠ sè ❤â❛ r ✈➔ p = r(u0 , v0 ) ∈ S ✳

Πp ❧➔ ♠➦t ♣❤➥♥❣ t✐➳♣ ①ó❝ ✈î✐ S t↕✐ p✳ ●✐↔ sû αp = (p, →
α ) ∈ Πp ❧➔

❈❤♦
●å✐




◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
♠ët ✈❡❝tì ❝â ❣è❝ t↕✐

p

✈➔

α ∈ Πp ✳

❑❤✐ ✤â tç♥ t↕✐

a, b ∈ R

s❛♦ ❝❤♦

α = aru (u0 , v0 ) + brv (u0 , v0 ).
❳➨t ✤÷í♥❣ ❝♦♥❣

γ : I = (− , ) → S

①→❝ ✤à♥❤ ❜ð✐

γ(t) = r(u0 + at, v0 + bt)
tr♦♥❣ ✤â

t∈I

t❤ä❛ ♠➣♥

(u0 + at, v0 + bt) ⊂ U ✳

❑❤✐ ✤â✱ ❞➵ t❤➜②


α.
γ(0) = p, γ (0) = →
◆❣÷í✐ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ t➟♣ ❤ñ♣
✈❡❝tì tr➯♥

R✳

{αp }

❑❤æ♥❣ ❣✐❛♥ ♥➔② ❦➼ ❤✐➺✉ ❧➔

❧➟♣ t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥

Tp S ✳

✭❑❤æ♥❣ ❣✐❛♥ t✐➳♣ ①ó❝ tr➯♥ ♠↔♥❤ ❤➻♥❤ ❤å❝✮✳
❑❤æ♥❣ ❣✐❛♥ TpS ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➳♣ ①ó❝ ❤➻♥❤ ❤å❝ ❝õ❛ S t↕✐ p✳
▼é✐ ✈❡❝tì αp ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✈❡❝tì t✐➳♣ ①ó❝ ❤➻♥❤ ❤å❝ ❝õ❛ S t↕✐ p✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

❚✐➳♣ t❤❡♦✱ t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✤❛ t↕♣ ✲ ❦❤→✐ ♥✐➺♠ trå♥❣ t➙♠ ❝õ❛
❧þ t❤✉②➳t ♠➦t tr♦♥❣

E3 ✳

✭✣❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3✮✳
❚➟♣ ❝♦♥ ❦❤æ♥❣ ré♥❣ S ❝õ❛ E3 ❣å✐ ❧➔ ♠ët ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3
♥➳✉ ♠é✐ p ∈ S ❝â ❧➙♥ ❝➟♥ ♠ð ✭❝õ❛ p tr♦♥❣ S ✮ ❧➔ ♠ët ♠↔♥❤ ❤➻♥❤ ❤å❝✳
▼é✐ t❤❛♠ sè ❤â❛ ❝õ❛ ♠↔♥❤ ❤➻♥❤ ❤å❝ ♥➔② ❣å✐ ❧➔ ♠ët t❤❛♠ sè ❤â❛ ✤à❛
♣❤÷ì♥❣ ❝õ❛ S ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳

❚❛ ✤÷❛ r❛ ♠ët ✈➼ ❞ö ✈➲ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉✳




◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❱➼ ❞ö ✶✳✷✳✷✳

❈❤♦ S ❧➔ ♠➦t ❝➛✉ tr♦♥❣ E3 ✈î✐ t➙♠ O✱ ❜→♥ ❦➼♥❤ R > 0✱
S = {p ∈ E3

s❛♦ ❝❤♦



Op = R}.

❑❤✐ ✤â✱ S ❧➔ ♠ët ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ ✈➻ ♥â ❧➔ ❤ñ♣ ❝õ❛ ✻ ♠↔♥❤ ❤➻♥❤ ❤å❝
❧➔ ✤ç t❤à ❝õ❛ ❝→❝ ❤➔♠ sè
(x, y) → ± R2 − x2 − y 2 ,
(y, z) → ± R2 − y 2 − z 2 ,
(z, x) → ± R2 − z 2 − x2 .
❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ❦❤↔ ✈✐ ✈➔ →♥❤
①↕ t✐➳♣ ①ó❝ tr➯♥ ❝→❝ ✤❛ t↕♣✳

✭⑩♥❤ ①↕ ❦❤↔ ✈✐ ❣✐ú❛ ❝→❝ ♠➦t✮✳
❈❤♦ S1✱ S2 ❧➔ ❤❛✐ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3✳ ⑩♥❤ ①↕ f : S1 −→ S2
✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ♥➳✉ f ❧✐➯♥ tö❝ ✈î✐ ♠å✐ t❤❛♠ sè ❤â❛ ✤à❛ ♣❤÷ì♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳

r1 : U1 −→ S1 ; r2 : U2 −→ S2

♠➔ f (r1(U1)) ⊂ r2(U2) t❤➻ →♥❤ ①↕ r2−1 ◦ f ◦ r1 : U1 −→ U2 ❧➔ ❦❤↔ ✈✐✳
✣➸ ♠✐♥❤ ❤å❛ ❦❤→✐ ♥✐➺♠ ♥➔②✱ t❛ ①➨t ✈➼ ❞ö s❛✉✳

❱➼ ❞ö ✶✳✷✳✸✳ ❈❤♦ S

❧➔ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3✳ ❑❤✐ ✤â✱ →♥❤ ①↕ ✤ç♥❣
♥❤➜t idS : S → S ❧➔ →♥❤ ①↕ ❦❤↔ ✈✐✳
❚❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ t✐➳♣ ①ó❝✳

❈❤♦ S1✱ S2 ❧➔ ✤❛ t↕♣ ❤❛✐ ❝❤✐➲✉ tr♦♥❣ E3 ✈➔ ❝❤♦
❧➔ →♥❤ ①↕ ❦❤↔ ✈✐✳ ❑❤✐ ✤â✱ f ❝↔♠ s✐♥❤ ♠ët →♥❤ ①↕

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳
f : S1 −→ S2




◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
Tp f : Tp S1 −→ Tp S2 ✤÷ñ❝ ①→❝

αp = (p, →
α )✱ ❣✐↔ sû γ : J −→ S1

✤à♥❤ ♥❤÷ s❛✉✿ ❱î✐ ♠é✐ αp ∈


♠➔ γ(t0) = p❀ →
γ (t0 ) = →
α t❤➻

Tp S1 ✱

Tp f (αp ) = (f ◦ p) (t0 ).
❈✉è✐ ❝ò♥❣✱ t❛ ❣✐î✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ♠➦t ❝❤➼♥❤ q✉②✳

▼ët t➟♣ ❝♦♥ M ⊂ Rn ✤÷ñ❝ ❣å✐ ❧➔ ♠➦t ❝❤➼♥❤ q✉②
♥➳✉ ✈î✐ ♠é✐ ✤✐➸♠ p ∈ M tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥ V ❝õ❛ p ✈➔ →♥❤ ①↕
x : U → Rn tø ♠ët t➟♣ ♠ð U ⊂ R2 ✤➳♥ V ∩ M s❛♦ ❝❤♦✿
✣à♥❤ ♥❣❤➽❛ ✶✳✾✳

✭✐✮✳ x : U → M ❧➔ ♠ët ♠↔♥❤ ❝❤➼♥❤ q✉②❀
✭✐✐✮✳ x : U → V ∩ M ❧➔ ♠ët ✤ç♥❣ ♣❤æ✐✱ ✈➻ ✈➟② x ❝â →♥❤ ①↕ ♥❣÷ñ❝ ❧✐➯♥
tö❝ x−1 : V ∩ M → U s❛♦ ❝❤♦ x−1 ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ ♠ët →♥❤ ①↕
❧✐➯♥ tö❝ F : W → Rn✱ ð ✤➙② W ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Rn✱ W ❝❤ù❛
V ∩ M✳ ▼é✐ →♥❤ ①↕ x : U → M ❣å✐ ❧➔ ♠ët ❜✐➸✉ ✤ç ✤à❛ ♣❤÷ì♥❣
❤❛② ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ p ∈ M✳

✶✵


❈❤÷ì♥❣ ✷
❚➼♥❤ ✤à♥❤ ❤÷î♥❣ ❝õ❛ ♠➦t
✷✳✶ ▼➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣
◆➳✉

V

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❤❛✐ ❝❤✐➲✉✱ t❛ ❣å✐ →♥❤ ①↕ t✉②➳♥ t➼♥❤

J : V → V

s❛♦ ❝❤♦

J 2 = −1

❧➔ ♠ët ❝➜✉ tró❝ ♣❤ù❝ tr➯♥

V✳

❱➻ t➜t

❝↔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❤❛✐ ❝❤✐➲✉ ✤➲✉ ✤➥♥❣ ❝➜✉ ♥➯♥ ♠é✐ ❦❤æ♥❣ ❣✐❛♥
t✐➳♣ ①ó❝

Mp

✈î✐ ♠ët ♠➦t ❝❤➼♥❤ q✉②

M

♥❤➟♥ ♠ët ❝➜✉ tró❝ ♣❤ù❝

Jp : Mp → Mp ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ▼ët ♠➦t ❝❤➼♥❤ q✉② M ⊂ R3

✤÷ñ❝ ❣å✐ ❧➔ ✤à♥❤ ❤÷î♥❣
♥➳✉ ♠é✐ ❦❤æ♥❣ ❣✐❛♥ t✐➳♣ ①ó❝ Mp ❝â ♠ët ❝➜✉ tró❝ ♣❤ù❝
Jp : Mp → Mp
p → Jp

❧➔ ♠ët ❤➔♠ sè ❧✐➯♥ tö❝✳ ▼ët ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ M ⊂ R3 ❧➔
♠ët ♠➦t ❝❤➼♥❤ q✉② ❝â t❤➸ ✤à♥❤ ❤÷î♥❣ ❝ò♥❣ ✈î✐ ♠ët ❝→❝❤ ❝❤å♥ ❝➜✉ tró❝
♣❤ù❝ p → Jp✳
✣à♥❤ ❧þ ✷✳✶✳

▼ët ♠➦t ❝❤➼♥❤ q✉② M ⊂ R3 ❧➔ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝ ❦❤✐
✶✶


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✈➔ ❝❤➾ ❦❤✐ ❝â ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝ p → U(p) s❛♦ ❝❤♦ ✈î✐ ♠é✐ p ∈ M
t÷ì♥❣ ù♥❣ ✈î✐ ♠ët ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à U(p) ∈ M⊥p ✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû t❛ ❝â p → U(p)✳ ❑❤✐ ✤â ✈î✐ ♠é✐ p ∈ M t❛ ①→❝
✤à♥❤

Jp : Mp → Mp

❜ð✐

Jp vp = U(p) × vp
Jp

❉➵ t❤➜②

→♥❤ ①↕ tø

Mp

✈➔♦

Mp : p → Jp

✭✷✳✶✮

❧➔ ❧✐➯♥ tö❝✳ ❚❛ s✉② r❛

J 2p vp = U(p) × (U(p) × vp )
= (U(p) · vp )U(p) − (U(p) · U(p))vp
= −vp .
◆❣÷ñ❝ ❧↕✐✱ ♥➳✉ ❝❤♦ ♠ët ♠➦t ❝❤➼♥❤ q✉②
♣❤ù❝

p → Jp

U(p) ∈ R3p

❧✐➯♥ tö❝ ✤÷ñ❝ ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝ tr➯♥

✈î✐ ♠å✐ ✈❡❝tì
✈➔

U(p)

Jp vp ✳

vp

❱➻

vp

trü❝ ❣✐❛♦ ✈î✐

✈➔

❚❛ ✤à♥❤ ♥❣❤➽❛

❑❤✐ ✤â

Jp vp

Mp ✳

❑❤✐ ✤â

✭✷✳✷✮

U(p)

t↕♦ t❤➔♥❤ ♠ët ❝ì sð tr♦♥❣

trü❝ ❣✐❛♦ ✈î✐ ❝↔

Mp

♥➯♥ s✉② r❛

Mp ✳
U(p)

t❤✉ë❝ ✈➔♦ ❝→❝❤ ❝❤å♥

Mp ✳

M✳

vp × Jp vp
vp × Jp vp

❦❤→❝ ❦❤æ♥❣✱ t❤✉ë❝

✣➸ ❝❤➾ r❛ r➡♥❣

tr♦♥❣

✈î✐ ♠ët ❝➜✉ tró❝

❜ð✐

U(p) =

vp

M ⊂ R3

vp ✱

①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✷✳✷✮ ❧➔ ❦❤æ♥❣ ♣❤ö
t❛ ❧➜②

wp

❧➔ ✈❡❝tì t✐➳♣ ①ó❝ ❦❤→❝ ✈❡❝tì✲❦❤æ♥❣

wp = avp + bJp vp

❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛

vp ✱

✈➔

wp × Jp wp = (avp + bJp vp ) × (−bvp + aJp vp ) = (a2 + b2 )(vp × Jp vp ).

✶✷


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❉♦ ✤â

(a2 + b2 )(vp × Jp vp )
vp × Jp vp
wp × Jp wp
=
=
,
wp × Jp wp
(a2 + b2 )(vp × Jp vp )
vp × Jp vp
✈➔ ✭✷✳✷✮ ❧➔ rã r➔♥❣✳ ❱➻

p → Jp

❧➔ ❧✐➯♥ tö❝ ♥➯♥

p → U(p)

❝ô♥❣ ❧➔ →♥❤

①↕ ❧✐➯♥ tö❝✳
✣à♥❤ ❧➼ ✭✷✳✶✮ ❝❤♦ ♣❤➨♣ ❝❤ó♥❣ t❛ ①→❝ ✤à♥❤ →♥❤ ①↕ ●❛✉ss ❝õ❛ ♠ët
♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ tò② þ tr♦♥❣

R3 ✳

❚❛ ❝â ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕

●❛✉ss ♥❤÷ s❛✉✳

❈❤♦ M ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ tr♦♥❣ R3
✈➔ ❝❤♦ U ❧➔ ♠ët tr÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à tr➯♥ M ①→❝ ✤à♥❤
❤÷î♥❣ ❝õ❛ M✳ ❑➼ ❤✐➺✉ S 2(1) ❧➔ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ R3✳ ❑❤✐ ✤â U
✤÷ñ❝ ①❡♠ ♥❤÷ ❧➔ ♠ët →♥❤ ①↕
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳

U : M → S 2 (1),

✈➔ ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ●❛✉ss ❝õ❛ M✳
✣➸ ❤✐➸✉ rã ❤ì♥ ✈➲ ✤à♥❤ ♥❣❤➽❛ ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣✱ t❛ ①➨t ❝→❝
❜ê ✤➲ s❛✉✳

❈❤♦ U ⊆ R2 ❧➔ ♠ët t➟♣ ♠ð✳ ✣ç t❤à Mh ❝õ❛ ❤➔♠ sè ❦❤↔
✈✐ ❧✐➯♥ tö❝ h : U → R ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳
❇ê ✤➲ ✷✳✶✳

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ♠↔♥❤ ▼♦♥❣❡ x : U → Mh ①→❝ ✤à♥❤ ❜ð✐
x(u, v) = (u, v, h(u, v)).
❑❤✐ ✤â

x

❜❛♦ ♣❤õ t♦➔♥ ❜ë

Mh ✱

♥❣❤➽❛ ❧➔

✶✸

x(U) = Mh ✳

❍ì♥ ♥ú❛✱

x

❧➔


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❝❤➼♥❤ q✉② ✈➔ ✤ì♥ →♥❤✳ ❱❡❝tì ♣❤→♣ t✉②➳♥ ❝õ❛ ♠➦t

U

ð

Mh

①→❝ ✤à♥❤

❜ð✐

U◦x=
❱❡❝tì ✤ì♥ ✈à
t❤➻

Mh

U

(−hu , −hv , 1)
xu × xv
=
.
xu × xv
1 + h2u + h2v

❦❤→❝ ❦❤æ♥❣ ð ♠å✐ ✤✐➸♠ tr➯♥

Mh

✈➔ t❤❡♦ ✣à♥❤ ❧➼ ✷✳✶

❧➔ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳

❇➜t ❦➻ ♠➦t M ⊂ R3 ♠➔ ❧➔ ✈➳t ❝õ❛ ♠ët ♠↔♥❤ ❝❤➼♥❤ q✉②
✤ì♥ →♥❤ x ✤➲✉ ❧➔ ♠➦t ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳

❇ê ✤➲ ✷✳✷✳

❍➻♥❤ ✷✳✶✿ ❍➻♥❤ ↔♥❤ ●❛✉ss ❝õ❛ ♠ët ♣❤➛♥ ♠➦t ①✉②➳♥
❍➻♥❤ ✷✳✶ ❧➔ ♠ët ✈➼ ❞ö ❝❤➾ r❛ r➡♥❣ →♥❤ ①↕ ●❛✉ss ❝õ❛ ♠ët ♣❤➛♥ t÷
♠➦t ①✉②➳♥ ❜❛♦ ♣❤õ ♠ët ♥û❛ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à✳ ✣➙② ❧➔ ♠ët sü tê♥❣ q✉→t
❤â❛ ❦❤→❝ ❝õ❛ ❇ê ✤➲ ✷✳✶✳

❇ê ✤➲ ✷✳✸✳ ❈❤♦ g : R3 → R ❧➔ ♠ët ❤➔♠ sè ❦❤↔ ✈✐ ✈➔ c ❧➔ ♠ët sè s❛♦ ❝❤♦

❦❤→❝ ❦❤æ♥❣ ✈î✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ M(c) = {p ∈ R3 | g(p) = c}✳
❑❤✐ ✤â M(c) ❝ô♥❣ ♥❤÷ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ M(c) ❧➔ ✤à♥❤
❤÷î♥❣ ✤÷ñ❝✳
grad g

❈❤ù♥❣ ♠✐♥❤✳ ❚❛
grad g

❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝

❧✉æ♥ trü❝ ❣✐❛♦ ✈î✐

M(c)

❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉② ✈➔

M(c)✳ ❚❛ ❣✐↔ sû r➡♥❣ grad g

✶✹

❦❤→❝ ❦❤æ♥❣ t↕✐


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
♠å✐ ✤✐➸♠ tr♦♥❣

M(c)✳

❚ø ✤â t❤➜② r➡♥❣

grad g
grad g
❧➔ ♠ët tr÷í♥❣ ✈❡❝tì ✤ì♥ ✈à ♠➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ð t➜t ❝↔ ❝→❝ ✤✐➸♠ t❤✉ë❝

M(c)

✈➔ ❧✉æ♥ trü❝ ❣✐❛♦ ✈î✐

M(c)✳

❑❤✐ ✤â✱ t❤❡♦ ✣à♥❤ ❧➼ ✷✳✶ t❤➻

M(c)

✈➔ ♠é✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ ♥â ❧➔ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳

✷✳✷ ❚r÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥
❝ö❝
❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣ ♠ët t➟♣ ❝♦♥

X

❝õ❛

Rn

✤÷ñ❝ ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ ♥â ❝❤ù❛

t➜t ❝↔ ❝→❝ ✤✐➸♠ tî✐ ❤↕♥ ❝õ❛ ♥â✳ ▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣✱
❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♣❤➛♥ ❜ò ❝õ❛ ♥â
▼é✐ t➟♣ ❝♦♥

X = X1 ∪ X2

X

❝õ❛

Rn \X

X

❧➔ t➟♣ ✤â♥❣

❧➔ ♠ð✳

✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥ t❤æ♥❣ ♥➳✉ ❜➜t ❦➻ sü ♣❤➙♥ t➼❝❤

X

t❤➔♥❤ ❝→❝ t➟♣ ❝♦♥ ✤â♥❣

t➟♣ ré♥❣ ♣❤↔✐ ❧➔ t➛♠ t❤÷í♥❣✱ ♥❣❤➽❛ ❧➔

X = X1

X1 , X2
❤♦➦❝

✈î✐

X1 ∩ X 2

❧➔

X = X2 ✳

❇ê ✤➲ ✷✳✹✳ ❈❤♦ M ❧➔ ♠ët ♠➦t ❝♦♥❣ ❝❤➼♥❤ q✉② ❧✐➯♥ t❤æ♥❣ ✤à♥❤ ❤÷î♥❣

✤÷ñ❝ tr♦♥❣ R3✳ ❑❤✐ ✤â M ❝â ❝❤➼♥❤ ①→❝ ❤❛✐ tr÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥
✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝✳
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ M ❧➔ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝ ♥➯♥ ❝â ➼t ♥❤➜t ♠ët tr÷í♥❣
✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝


p → v(p)

❝ö❝ ❦❤→❝ tr➯♥

p → U(p)

tr➯♥

M

✳ ●✐↔

❧➔ ♠ët tr÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥

M✳

❈→❝ t➟♣

W± = {p ∈ M | U(p) = ±v(p)}

✶✺


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❧➔ ❝→❝ t➟♣ ✤â♥❣ ❜ð✐ ✈➻
❍ì♥ ♥ú❛

U

✈➔

v

M = W+ ∪ W− ✳

trò♥❣ ✈î✐

W+

✤à♥❤ tr➯♥

M

❤♦➦❝

❧➔

W− ✳

❧➔ ❧✐➯♥ tö❝✳

❚ø t➼♥❤ ❧✐➯♥ t❤æ♥❣ ❝õ❛

M

t❛ s✉② r❛

M

❉♦ ✤â✱ tr÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥ ✤÷ñ❝ ①→❝

p → U(p)

✈➔

p → −U(p)✳

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❈❤♦ M ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ tr♦♥❣ R3 ✳

▼ët sü ✤à♥❤ ❤÷î♥❣ ❝õ❛ M ❧➔ ✈✐➺❝ ❝❤å♥ ♠ët tr÷í♥❣ ✈❡❝tì ♣❤→♣ t✉②➳♥
✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝ tr➯♥ M✳

✷✳✸ ❈→❝ ♠↔♥❤ ❝â ❝ò♥❣ ✤à♥❤ ❤÷î♥❣
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t✱ t❛ ❦❤æ♥❣ t❤➸ ♣❤õ ♠ët ♠➦t ❝❤➼♥❤ q✉②
❜ð✐ ♠ët ♠↔♥❤ ✈→ ✤ì♥✳ ◆➳✉ ❝❤ó♥❣ t❛ ❝â ♠ët ❤å ❝→❝ ♠↔♥❤ ✈→✱ ❝❤ó♥❣
t❛ ❝➛♥ ♣❤↔✐ ❜✐➳t ❝→❝ sü ✤à♥❤ ❤÷î♥❣ ❝õ❛ ❝→❝ ♠↔♥❤ ✈→ ✤÷ñ❝ ❧✐➯♥ ❤➺ ✈î✐
♥❤❛✉ ♥❤÷ t❤➳ ♥➔♦ ✤➸ ✤÷❛ r❛ ✤à♥❤ ❤÷î♥❣ ❝õ❛ ♠ët ♠➦t ❝❤➼♥❤ q✉②✳ ❉♦
✈➟② t❛ ❝➛♥ ❜ê ✤➲ s❛✉✳

❈❤♦ M ⊂ R3 ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉②✱ U ✱ V ❧➔ ❝→❝ t➟♣ ♠ð
tr♦♥❣ R2 ✳ ◆➳✉ x : U → M ✈➔ y : V → M ❧➔ ❝→❝ ♠↔♥❤ ✈→ s❛♦ ❝❤♦
x(U) ∩ y(V) ❦❤→❝ ré♥❣✱ t❤➻
❇ê ✤➲ ✷✳✺✳

yu × yv = det(J (x−1 ◦ y))xu × xv ,

tr♦♥❣ ✤â J (x−1 ◦ y) ❧➔ ♠❛ tr➟♥ ❏❛❝♦❜✐❛♥ ❝õ❛ x−1 ◦ y✳ Ð ✤➙②✱ (u, v) ❧➔
❝→❝ tå❛ ✤ë tr➯♥ V ✱ ❝á♥ (u, v) ❧➔ ❝→❝ tå❛ ✤ë tr➯♥ U ✳

✶✻


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â
∂u
xu +
∂u

∂u

= det  ∂u
∂v
∂u

yu × yv =

∂v
∂v
∂u
xv ×
xu + xv
∂u
∂v
∂v

∂u
∂v 
x × xv
∂v  u
∂v

= det(J (x−1 ◦ y))xu × xv

❚❛ ♥â✐ r➡♥❣ ❝→❝ ♠↔♥❤ x : U → M ✈➔ y : V → M
tr➯♥ ♠ët ♠➦t ❝❤➼♥❤ q✉② M ✈î✐ x(U) ∩ y(V) ❦❤→❝ ré♥❣ ❝â ❝ò♥❣ ✤à♥❤
❤÷î♥❣ ♥➳✉ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ❏❛❝♦❜✐❛♥ J (x−1 ◦ y) ❧➔ ❞÷ì♥❣ tr♦♥❣
x(U) ∩ y(V)✳

✣à♥❤ ♥❣❤➽❛ ✷✳✹✳

✣à♥❤ ❧þ ✷✳✷✳ ▼ët ♠➦t ❝❤➼♥❤ q✉② M ⊂ R3

❧➔ ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝ ♥➳✉ ✈➔
❝❤➾ ♥➳✉ ❝â t❤➸ ❜❛♦ ♣❤õ M ❜ð✐ ♠ët ❤å B ❝→❝ ♠↔♥❤ ✤ì♥ →♥❤ ❝❤➼♥❤ q✉②
s❛♦ ❝❤♦ ❜➜t ❦➻ ❤❛✐ ♠↔♥❤ (x, U)✱ (y, V) ✈î✐ x(U) ∩ y(V) = ∅ ❝â ❝ò♥❣
✤à♥❤ ❤÷î♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉② ✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳ ❚ø
✣à♥❤ ❧➼ ✷✳✶✱
▲➜②

M ❝â ♠ët ✈❡❝tì ♣❤→♣ t✉②➳♥ ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝ p → U(p)✳

U ❧➔ ♠ët ❤å ❝→❝ ♠↔♥❤ ✤ì♥ →♥❤ ❝❤➼♥❤ q✉② ♠➔ ❤ñ♣ ❝õ❛ ❝❤ó♥❣ ❝❤ù❛

M✳
❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ ♠é✐ ♠↔♥❤
tr♦♥❣

U

❧➔ ❧✐➯♥ t❤æ♥❣✳ ❚❛ ♣❤↔✐ ①➙② ❞ü♥❣ tø

❝ò♥❣ ✤à♥❤ ❤÷î♥❣ ♠➔ ❤ñ♣ ❝õ❛ ❝❤ó♥❣ ❝❤ù❛

U

♠ët ❤å

❝→❝ ♠↔♥❤ ❝â

x

❧➔ ♠↔♥❤ ✤ì♥

M✳

✣➸ ❧➔♠ ✤✐➲✉ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ t❛ ❧÷✉ þ r➡♥❣ ♥➳✉

✶✼

B


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
→♥❤ ❝❤➼♥❤ q✉② ❜➜t ❦➻ tr♦♥❣

M

t❤➻

x

✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

x(u, v) = x(v, u)
M✱ x

❝ô♥❣ ❧➔ ♠ët ♠↔♥❤ ✤ì♥ →♥❤ ❝❤➼♥❤ q✉② tr♦♥❣
♥❤❛✉ ✈➻

U

x

♥❣÷ñ❝ ❤÷î♥❣

◆➳✉

x

❧➔ ♠ët ♠↔♥❤

xu × xv = −xu × xv ✳

❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤å♥ ❝→❝ ♠↔♥❤ t❤✉ë❝ ❤å
tr♦♥❣

✈➔

B✳

✈➔

xu × xv
=U
xy × xv
t❛ ✤➦t

x

✈➔♦ tr♦♥❣

B✱

❝á♥ ♥➳✉

xu × xv
= −U
xy × xv
t❤➻ t❛ ✤➦t

x

✈➔♦

❤÷î♥❣ ♠➔ ❝❤ù❛

B✳

x

B

❧➔ ♠ët ❤å ❝→❝ ♠↔♥❤ ❝ò♥❣ ✤÷ñ❝ ✤à♥❤

M✳

◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû
❤÷î♥❣✳ ◆➳✉

❑❤✐ ✤â

M

❝❤ù❛ tr♦♥❣ ❤å

❧➔ ♠ët ♠↔♥❤ t❤✉ë❝ ❤å

❝❤ó♥❣ t❛ ①→❝ ✤à♥❤

Ux

tr➯♥

x(U)

y : V → M

B

❝→❝ ♠↔♥❤ ❝ò♥❣ ✤÷ñ❝ ✤à♥❤

✤÷ñ❝ ①→❝ ✤à♥❤ tr➯♥

U ⊂ R2 ✱

❜ð✐

Ux (x(u, v)) =
▲➜②

B

xu × xv
(u, v).
xu × xv

❧➔ ♠ët ♠↔♥❤ ❦❤→❝ tr♦♥❣

✶✽

B

✈î✐

x(U) ∩ y(V) = ∅✳


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝
❚❤❡♦ ❇ê ✤➲ ✷✳✺ tr➯♥

x(U) ∩ y(V)

t❛ ❝â

yu × yv
(u, v)
y y × yv
detJ (x−1 ◦ y) xu × xv
=
|detJ (x−1 ◦ y)| xu × xv
x u × xv
=
(u, v)
x u × xv

Uy (y(u, v)) =

(u, v)

= Ux (x(u, v)).
❉♦ ✤â✱ t❛ ①→❝ ✤à♥❤ ✤÷ñ❝ ♣❤→♣ t✉②➳♥
tr➯♥

x(U)

❝❤♦ ❜➜t ❦➻ ♠↔♥❤

x

tr♦♥❣

U ❝õ❛ M ❜➡♥❣ ❝→❝❤ ✤➦t U = Ux
B✳

❚❤❡♦ ✣à♥❤ ❧➼ ✷✳✶✱

M

❧➔ ♠➦t

✤à♥❤ ❤÷î♥❣ ✤÷ñ❝✳
❚ø ✣à♥❤ ❧➼ ✷✳✷ t❛ t❤✉ ✤÷ñ❝ ❤➺ q✉↔ s❛✉✳

▼ët ❤å ❝→❝ ♠↔♥❤ ✤ì♥ →♥❤ ❝❤➼♥❤ q✉② ❝ò♥❣ ✤÷ñ❝ ✤à♥❤
❤÷î♥❣ tr➯♥ ♠ët ♠➦t ♣❤➥♥❣ ❝❤➼♥❤ q✉② R3 ①→❝ ✤à♥❤ ♠ët tr÷í♥❣ ✈❡❝tì
♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ①→❝ ✤à♥❤ t♦➔♥ ❝ö❝ ❝õ❛ ♠➦t M tr♦♥❣ R3✱ tù❝ ❧➔ ①→❝
✤à♥❤ ♠ët ❤÷î♥❣ ❝õ❛ M✳
❍➺ q✉↔ ✷✳✶✳

❚➟♣ ↔♥❤ q✉❛ →♥❤ ①↕ ●❛✉ss ❝õ❛ ♠✐➲♥ ①➼❝❤ ✤↕♦

hyperboloid1[a, b, c](u, v) = (a cosh v cos u, b cosh v sin u, c sinh v),
{hyperboloid1[1, 1, 1](u, v) | 0 ≤ u ≤ 2π, −1 ≤ v ≤ 1}
❝õ❛ ♠ët ❤②♣❡❜♦❧♦✐❞ ♠ët t➛♥❣ ✤÷ñ❝ ♠✐♥❤ ❤å❛ ð ❍➻♥❤ ✷✳✷✳ ❚➟♣ ↔♥❤
t♦➔♥ ❜ë ❤②♣❡❜♦❧♦✐❞ ❝ô♥❣ ❧➔ ♠ët ♠✐➲♥ ①➼❝❤ ✤↕♦ ❜à ❝❤➦♥ ✈➻ ❝→❝ ♣❤→♣
t✉②➳♥ ❝õ❛ ♠➦t t✐➳♥ tî✐ ❝→❝ ♣❤→♣ t✉②➳♥ ❝õ❛ ♥â♥ t✐➺♠ ❝➟♥✳

✶✾


◆❣✉②➵♥ ❚❤à ⑩♥❤ ❈❤✐♥❤

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❚➟♣ ↔♥❤ ❝õ❛ ❤②♣❡❜♦❧♦✐❞ ❤❛✐ t➛♥❣ q✉❛ →♥❤ ①↕ ●❛✉ss ❣ç♠ ❝â ❤❛✐ ✤➽❛
✤è✐ ❝ü❝✳

❍➻♥❤ ✷✳✷✿ ❍➻♥❤ ↔♥❤ ●❛✉ss ❝õ❛ ♠➦t ❤②♣❡r❜♦❧♦✐❞

✷✵


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