❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✣⑨ ◆➂◆●
❑❍❖❆ ❚❖⑩◆
◆●❯❨➍◆ ❚❍➚ P❍×Đ◆●

✣❸■ ❙➮ ❚❘×❮◆● ❱❊❈❚❖❘ ❱⑨ ❚➑❈❍ P❍❹◆ ❈❍Ĩ◆●

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
◆❣➔♥❤✿ ỷ

ữợ ề ❉×❒◆●

✣➔ ◆➤♥❣✱ ✺✴✷✵✶✹


▼ö❝ ❧ö❝
▲❮■ ❈❷▼ ❒◆✦
▼Ð ✣❺❯
✶ ◆❍Ú◆● ❑❍⑩■ ◆■➏▼ ❈❒ ❇❷◆

✷
✸
✺

✶✳✶ ❇❛♦ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷ ✣↕✐ sè ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✸ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✶✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✶✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ t✉②➳♥ t➼♥❤ t❤✉➛♥
♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✸✳✸ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ t✉②➳♥ t➼♥❤ ❦❤æ♥❣
t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✹ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✺ ✣÷í♥❣ ❝♦♥❣ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ✣❸■ ❙➮ ❚❘×❮◆● ❱❊❈❚❖❘ ❱⑨ ❚➑❈❍ P❍❹◆ ❈❍Ĩ◆● ✶✽
✷✳✶
✷✳✷
✷✳✸
✷✳✹

❱➼ ❞ư ✈➲ ✤↕✐ sè ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤✉②➸♥ ✤ê✐ s❛♥❣ tå❛ ✤ë t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤➨♣ ❧➜② t➼❝❤ ♣❤➙♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳
❙û ❞ö♥❣ ❝→❝ ❣â✐ t♦→♥ ❤å❝ ▼❛♣❧❡ ✈➔ ▼❛t❤❈❛❞

❑➌❚ ▲❯❾◆
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳

✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✶✽
✷✻
✸✶
✹✶

✹✸
✹✹
✶


▲❮■ ❈❷▼ ❒◆✦
❑❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ồ ữ P ồ
ữợ sỹ ữợ t t ừ ổ ❚❤ị②

❉÷ì♥❣✳ ❊♠ ①✐♥ ❣û✐ ✤➳♥ ❝ỉ ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝✳
❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❦❤â❛ ❧✉➟♥✱ t❤æ♥❣ q✉❛ ❝→❝ ❜➔✐ ❣✐↔♥❣✱
❜➔✐ ❤å❝✱ ❡♠ ❧✉æ♥ ữủ sỹ q t ú ù ỳ ỵ ❦✐➳♥ ✤â♥❣
❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦✱ ❚❙✱❚❤❙ t❤✉ë❝ ❦❤♦❛ ❚♦→♥ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲
✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✳ ❚ø ✤→② ❧á♥❣ ♠➻♥❤✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥
t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❝ỉ✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠
✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥✱ q✉❛♥ t➙♠✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt
t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❦❤â❛ ❧✉➟♥✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱✳ ✳ ✳ t➜t ❝↔ ♥❤ú♥❣
♥❣÷í✐ ✤➣ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤
❦❤â❛ ❧✉➟♥ ♥➔②✳

✷


▼Ð ✣❺❯
❈→❝ ✤è✐ t÷đ♥❣ ❤➻♥❤ ❤å❝✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ✤÷í♥❣ ❝♦♥❣ ❤♦➦❝ ❜➲ ♠➦t✱ ♥❤÷ ✤➣
❜✐➳t✱ ❝â t❤➸ ✤÷đ❝ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✭ ♣❤÷ì♥❣ tr➻♥❤ t÷í♥❣ ♠✐♥❤ ❤♦➦❝
♣❤÷ì♥❣ tr➻♥❤ sỷ ử ỵ ữỡ tr ✈✐ ♣❤➙♥✱
❝❤ó♥❣ t❛ ❝â t❤➸ ❝✉♥❣ ❝➜♣ ♥❤ú♥❣ ❝→❝❤ ❦❤→❝✳ ử õ ởt trữớ ữợ
tữỡ ữỡ õ ởt ❤å ❝→❝ ✤÷í♥❣ ❝♦♥❣✮ tr♦♥❣ ♠ët ♠➦t ♣❤➥♥❣
♥➔♦ ✤â✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ✭❜➡♥❣ ❝→❝❤ trü❝ t✐➳♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ t❤÷í♥❣ t÷ì♥❣ ù♥❣✮ ♥❤ú♥❣ ✤÷í♥❣ ❝♦♥❣ t➼❝❤ ♣❤➙♥ ❝❤♦ tr÷í♥❣ ♥➔②✳
❇➜t ❦ý ✤÷í♥❣ ❝♦♥❣ ♥➔♦✱ ♥❤÷ ✤➣ ❜✐➳t✱ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ♥â ✤➲✉ ❝â ✈❡❝t♦r
t✐➳♣ t✉②➳♥✱ trũ ợ ữợ ừ ữớ trữợ t ✤â✳
❚÷ì♥❣ tü ♥❤÷ ✈➟② ✭♥❤÷♥❣ ❦❤â ❦❤➠♥ ❤ì♥✮✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ①➙② ❞ü♥❣
♠ët ❜➲ ♠➦t tr♦♥❣ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤❛ ❝❤✐➲✉✱ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ❜➲ ♠➦t
♠➦t ♣❤➥♥❣ t✐➳♣ t trũ ợ t tữỡ ự tr ồ
trữợ tr ợ ỵ ờ t sỹ tỗ t

t t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝❤ó♥❣ t❛ ♣❤↔✐
❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ✈➔ ❞♦ õ ố õ ợ
ỵ ự t ỡ
✈➟②✱ ①✉➜t ❤✐➺♥ ❝→❝ ✤↕✐ sè ❝õ❛ tr÷í♥❣ ✈❡❝t♦r✱ ✤â ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥
t➼♥❤ ✈ỵ✐ ❝➜✉ tró❝ ❜ê s✉♥❣✳ ❈→❝ ✤↕✐ sè ♥❤÷ ✈➟② ❝â t❤➸ ❧➜② t➼❝❤ ♣❤➙♥ ✈➔
t❤✉ ữủ ỳ t tữỡ ự ợ số ✤â✳
❚r♦♥❣ ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔② ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❧✐➯♥ q
ự t ỗ t tr ổ ❣✐❛♥ ♣❤ù❝ ✸ ❝❤✐➲✉✳
✸


❈ư t❤➸ ❧➔✱ ❝❤♦ ♠ët ✤↕✐ sè tr÷í♥❣ ✈❡❝t♦r t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥
C3 ✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ①➙② ỹ ữủ s tỹ ỗ t s
q✉❛ ❣è❝ tå❛ ✤ë ✈➔ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ♥â õ sỹ t ú ợ trữớ
t ừ số ✤➣ ❝❤♦✳
▼ët ❝→❝❤ tü ♥❤✐➯♥✱ ❝â t❤➸ ♥â✐ r➡♥❣ ❝❤ó♥❣ trữớ t ú ợ
t ú t ✤❛♥❣ t❤↔♦ ❧✉➟♥✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ①➙② ❞ü♥❣ ❜➲ t
ỹ tr số trữớ t ú ợ ♠➦t✳
❇➲ ♠➦t ①➙② ❞ü♥❣ ✤÷đ❝ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❧➔ ♠ët tr ỳ s
ỗ t tr ổ C3
◆➤♥❣✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹
❙✐♥❤ ✈✐➯♥
◆❣✉②➵♥ ❚❤à P❤÷đ♥❣

✹


❈❤÷ì♥❣ ✶
◆❍Ú◆● ❑❍⑩■ ◆■➏▼ ❈❒ ❇❷◆
✶✳✶ ❇❛♦ t✉②➳♥ t➼♥❤

✣à♥❤ ♥❣❤➽❛ ✶✳ ●✐↔ sû ❆ ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ tr ổ

ổ tỗ t ởt ổ ❝õ❛ ❳ ❝❤ù❛ ❆✳ ●✐❛♦ ❝õ❛ ❤å t➜t ❝↔ ❝→❝
❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❝❤ù❛ ❆ ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝❤ù❛ ❆✳ ❑❤ỉ♥❣ ❣✐❛♥
❝♦♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ s✐♥❤ ❜ð✐ ❆ ❤❛② ❧➔ ❜❛♦ t✉②➳♥ t➼♥❤ ❝õ❛
❆✳
❈❤♦ x1, x2, . . . , xn ❧➔ ♥ ✈❡❝t♦r ✭♣❤➛♥ tû✮ ❝õ❛ ❚✲❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r ❳✳
❚❛ ❣å✐ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ✈❡❝t♦r x1, x2, . . . , xn ❧➔ ♠ët ✈❡❝t♦r
① ❝â ❞↕♥❣
x = α1 x1 + α2 x2 + . . . + αn xn , αi ∈ T, i = 1 . . . n

❑❤✐ ✤â t❛ ♥â✐ ✈❡❝t♦r ① ❜✐➸✉ ❞✐➵♥ t✉②➳♥ t➼♥❤ ✤÷đ❝ q✉❛ ❝→❝ ✈❡❝t♦r x1, x2, . . . , xn✳
❇❛♦ t✉②➳♥ t➼♥❤ ❝õ❛ t➟♣ ❆ ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛
❝→❝ ♣❤➛♥ tû t❤✉ë❝ ❆✳

✶✳✷ ✣↕✐ sè ▲✐❡
✣à♥❤ ♥❣❤➽❛ ✷✳ ❈❤♦ K ❧➔ ♠ët tr÷í♥❣ ✈➔ ▲ ❧➔ ♠ët K ✲ ❑●❱❚✳ ❚❛ ♥â✐
▲ ❧➔ ♠ët K ✲ ✤↕✐ sè ▲✐❡ ♥➳✉ ▲ ✤÷đ❝ tr❛♥❣ ❜à t❤➯♠ ♠ët ♣❤➨♣ ♥❤➙♥ ❣å✐ ❧➔

t➼❝❤ ▲✐❡ ✭❤❛② ♠â❝ ▲✐❡✮✳

✺


[., .] : L × L → L
(x, y) → [x, y]

ữủ ồ t ừ ợ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ s❛✉✿
✭✐✮ (L1) : [., .] s♦♥❣ t✉②➳♥ t➼♥❤✳
✭✐✐✮(L2) : [., .] ♣❤↔♥ ①ù♥❣ ✿ [x, x] = 0 ∀x ∈ ▲

✭✐✐✐✮(L3) : [., .] tọ ỗ t
[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0

❱➼ ❞ö ✶✳ ❈❤ù♥❣ r R3 ợ t t õ ữợ ❝õ❛ ❤➻♥❤
sì ❝➜♣ ❧➔ ♠ët ✤↕✐ sè ▲✐❡✳

❈❤ù♥❣ ♠✐♥❤✿
✐✮ (L1) : [., .] s♦♥❣ t✉②➳♥ t➼♥❤✳

❑

∀x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), z = (z1 , z2 , z3 ) ∈ R3 ✈➔ α, β ∈ ✳ ❚❛ ❝â✿
[αx + βy, z] = (αx + βy) × z = α(x × z) + β(y × z) = α[x, z] + β[y, z]
[x, αy + βz] = x × (αy + βz) = α(x × y) + β(x × z) = α[x, y] + β[x, z]

✐✐✮ (L2) : [., .] ♣❤↔♥ ①ù♥❣✳ ❚❤➟t ✈➟②✱ ∀x ∈ R3 t❤➻ [x, x] = x × x = 0✳
✐✐✐✮ (L3) : [., .] t❤ä❛ ♠➣♥ ỗ t
x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ), z = (z1 , z2 , z3 ) ∈ R3 ✳

❚❛ ❝â✿

[x, [y, z]] = x × (y × z)
[z, [x, y]] = z × (x × y)
[y, [z, x]] = y × (z × x)

❑❤✐ ✤â✱ [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0✳
❱➟② R3 ✈ỵ✐ t➼❝❤ ▲✐❡ ♥❤÷ tr➯♥ ❧➔ ♠ët ✤↕✐ sè ▲✐❡✳
◆❤➟♥ ①➨t ✶✳ • ❚r➯♥ ♠é✐ K ❜➜t ❦➻✱ ▲ ✤➲✉ ❝â t❤➸ tr❛♥❣ ❜à t➼❝❤ ▲✐❡ t➛♠
t❤÷í♥❣ [x, y] = 0 ∀x, y ∈ ▲ ✤➸ trð t❤➔♥❤ ✤↕✐ sè ▲✐❡✳ ❑❤✐ ✤â✱ t❛ ❣å✐ ▲ ❧➔
♠ët ✤↕✐ sè ▲✐❡ ❣✐❛♦ ❤♦→♥✳

• ❚r➯♥ ❝ị♥❣ ♠ët K ✲ ❑●❱❚ ▲ t❛ ❝â t❤➸ tr❛♥❣ ❜à ♥❤✐➲✉ ❤❛② ✈æ sè ✤↕✐ sè
▲✐❡ ❦❤→❝ ♥❤❛✉ ❦❤✐ t❤❛② ✤ê✐ ❝→❝ t➼❝❤ ▲✐❡ ❦❤→❝ ♥❤❛✉✳
✻


▼é✐ ✤↕✐ sè ▲✐❡ ❧➔ ♠é✐ ❑●❱❚ ♥➯♥ sè ❝❤✐➲✉ ừ số số
ừ
ã

Pữỡ tr ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣
❈â ♥❤✐➲✉ ❜➔✐ t♦→♥ ❝õ❛ t❤ü❝ t t ỵ tt tợ t➻♠ ♠ët
❤➔♠ z(x1, x2, . . . , xn)✱ ❜✐➳♥ x1, x2, . . . , xn✱ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤
∂z
∂z ∂ 2 z
∂kz
F (x1 , x2 , . . . , xn , z,
,...,
,
, . . . , k1
)=0
∂x1
∂xn ∂x21
∂x1 . . . ∂xknn

✣à♥❤ ♥❣❤➽❛ ✸✳ P❤÷ì♥❣ tr➻♥❤✱ tr♦♥❣ ✤â ❝â ➼t ♥❤➜t ♠ët ✤↕♦ ❤➔♠ r✐➯♥❣

❝➜♣ ❦ ❝õ❛ ❤➔♠ ❝❤÷❛ ❜✐➳t z(x1, x2, . . . , xn) ✈➔ ❦❤ỉ♥❣ ❝â ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣
❝➜♣ ❝❛♦ ❤ì♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❦ ❝õ❛
❝→❝ ❜✐➳♥ x1, x2, . . . , xn✳
❍➔♠ z(x1, x2, . . . , xn) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ tr♦♥❣ ♠ët ♠✐➲♥ ♥➔♦

✤â ❝õ❛ ❜✐➳♥ x1, x2, . . . , xn ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❤❛② t➼❝❤ ♣❤➙♥ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ tr➯♥ ♠✐➲♥ ✤â✳

❱➼ ❞ư ✷✳ P❤÷ì♥❣ tr➻♥❤

∂ 3u
∂ 2u
∂ 2u
∂u
+3
+ 2 + 5u
=0
3
∂x
∂x∂y ∂y
∂x

✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✸ ❝õ❛ ❤❛✐ ❜✐➳♥ ✭①✱②✮✳

❧➔ ♣❤÷ì♥❣ tr➻♥❤

✶✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ♠ët
P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣
F (x1 , x2 , . . . , xn , z,

∂z
∂z
,...,
)=0
∂x1

∂xn

P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ t✉②➳♥ t➼♥❤ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣
X1 (x1 , x2 , . . . , xn , z)

∂z
∂z
+ . . . + Xn (x1 , x2 , . . . , xn , z)
=
∂x1
∂xn
f (x1 , x2 , . . . , xn , z)

✼


ừ ữỡ tr tr ỗ t ✵✱ ❝á♥ ❝→❝ ❤➔♠
Xi (x1 , x2 , . . . , xn , z) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ③ t❤➻ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥
t➼♥❤ t❤✉➛♥ ♥❤➜t ✿
X1 (x1 , x2 , . . . , xn )

∂z
∂z
+ . . . + Xn (x1 , x2 , . . . , xn )
= 0.
∂x1
∂xn

∂u
∂u

+y
+z
= u ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠
❱➼ ❞ư ✸✳ P❤÷ì♥❣ tr➻♥❤ x ∂u
∂x
∂y
∂z

r✐➯♥❣ ❝➜♣ ✶ ❝õ❛ ❜❛ ❜✐➳♥ ✭①✱②✱③✮✳
∂u
P❤÷ì♥❣ tr➻♥❤ ( ∂u
)2 +
= 0 ❝ơ♥❣ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣
∂x
∂t
✶ ❝õ❛ ❤❛✐ ❜✐➳♥ ✭①✱t✮✳

✶✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ t✉②➳♥ t➼♥❤ t❤✉➛♥
♥❤➜t
❳➨t ♣❤÷ì♥❣ tr➻♥❤✿
X1 (x1 , x2 , . . . , xn )

∂z
∂z
+ . . . + Xn (x1 , x2 , . . . , xn )
= 0.
∂x1
∂xn

✭✶✳✶✮


●✐↔ sû r➡♥❣ Xi(x1, x2, . . . , xn) ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tư❝ ❝ị♥❣ ✈ỵ✐ ❝→❝ ✤↕♦ ❤➔♠
r✐➯♥❣ ❝➜♣ ♠ët ❝õ❛ ❝❤ó♥❣ t❤❡♦ t➜t ❝↔ ❝→❝ ❜✐➳♥ ð tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦
✤â ❝õ❛ ✤✐➸♠ (x01, x02, . . . , x0n)✳
❚❛ ♣❤↔✐ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✱ tù❝ ❧➔ t➻♠ ♠ët ❤➔♠
z(x1 , x2 , . . . , xn ) ①→❝ ✤à♥❤ ✈➔ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ (x01 , x02 , . . . , x0n )
s❛♦ ❝❤♦ ♥â t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ❧➙♥ ❝➟♥ ➜②✳
P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❝â ♥❣❤✐➺♠ ❤✐➸♥ ♥❤✐➯♥ ✭♥❣❤✐➺♠
t➛♠ t❤÷í♥❣✮ z = C ✱ tr♦♥❣ ✤â ❈ ❧➔ ❤➡♥❣ sè✳ ◆❣♦➔✐ r❛✱ t❛ s➩ ❝❤ù♥❣ tä r
õ õ ổ số ổ t tữớ
ũ ợ ữỡ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ t❛ ①➨t ❤➺ ♣❤÷ì♥❣
✽


tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤è✐ ①ù♥❣ s❛✉✿
dx1
dx2
dxn
=
= ... =
X1 (x1 , x2 , . . . , xn ) X2 (x1 , x2 , . . . , xn )
Xn (x1 , x2 , . . . , xn )

✭✶✳✷✮
✣à♥❤ ❧➼ ✶✳ ◆➳✉ ϕ(x1, x2, . . . , xn) ❧➔ t➼❝❤ ♣❤➙♥ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝õ❛ ❤➺ (1.2)✱
tù❝ ❧➔
dϕ|(2) =

∂ϕ
∂ϕ

∂ϕ
X1 +
X2 + . . . +
Xn ≡ 0
∂x1
∂x2
∂xn

tr♦♥❣ ♠ët ♠✐➲♥ ♥➔♦ ✤â ❝õ❛ ❜✐➳♥ sè x1, x2, . . . , xn✱ t❤➻ ❤➔♠ sè
z = ϕ(x1 , x2 , . . . , xn ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1.1)✳
✣↔♦ ❧↕✐✱ ♥➳✉ z = ψ(x1, x2, . . . , xn) ❧➔ ♥❣❤✐➺♠ ❝õ❛ (1.1)✱ t❤➻ ψ(x1, x2, . . . , xn)
❧➔ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➺ (1.2)✳
❚ø ✤à♥❤ ❧➼ tr➯♥ t❛ s✉② r❛ r➡♥❣✱ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠
r✐➯♥❣ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t (1.2) tữỡ ữỡ ợ ữỡ tr
tữớ ố ①ù♥❣ (1.2)✳ ◆➳✉ ❜✐➳t(n − 1) t➼❝❤ ♣❤➙♥ ✤ë❝ ❧➟♣✱
ϕ1 (x1 , x2 , . . . , xn )✱ ϕ2 (x1 , x2 , . . . , xn )✱ ϕn−1 (x1 , x2 , . . . , xn ) ❝õ❛ ❤➺ (1.2) t❤➻
♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ (1.1) s➩ ❧➔
z = Φ(ϕ1 , . . . , ϕn−1 ).
∂u
∂u
❱➼ ❞ö ✹✳ ▼✉è♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ x ∂u
+y
+z
= 0 t ú ỵ r
x
y
z
dy
dz
=

= õ
ữỡ tr ✤è✐ ①ù♥❣ t÷ì♥❣ ù♥❣ ❝â ❞↕♥❣ dx
x
y
z
y
z
✷ t➼❝❤ ♣❤➙♥ ✤ë❝ ❧➟♣ ❧➔ ϕ1 = x , ϕ2 = x ❝❤ù♥❣ tä ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤➣ ❝❤♦ ❧➔ u = Φ( xy , xz )✳

✾


✶✳✸✳✸ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣
t❤✉➛♥ ♥❤➜t
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t
X1 (x1 , x2 , . . . , xn , z)

∂z
∂z
+. . .+Xn (x1 , x2 , . . . , xn , z)
= f (x1 , x2 , . . . , xn , z)
∂x1
∂xn

✭✶✳✸✮
tr♦♥❣ ✤â ❝→❝ ❤➔♠ sè Xi ✈➔ f ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tư❝ ❝ị♥❣ ✈ỵ✐ ✤↕♦ ❤➔♠
r✐➯♥❣ ❝➜♣ ♠ët ❝õ❛ ❝❤ó♥❣ t❤❡♦ t➜t ❝↔ ❝→❝ ❜✐➳♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â
❝õ❛ ✤✐➸♠ (x01, x02, . . . , x0n, z0)✳ ❚❛ s➩ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣
t❤✉➛♥ ♥❤➜t ❜➡♥❣ ❝→❝❤ ✤÷❛ ✈➲ ❤➺ t❤✉➛♥ t ừ ữỡ tr

s ữủ t ữợ ♠ët ❤➔♠ ➞♥✿
U (x1 , x2 , . . . , xn , z) = 0✱
tr♦♥❣ ✤â U ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ t❤❡♦ t➜t ❝↔ ❝→❝ ✤è✐ sè ✈➔ t❤ä❛ ♠➣♥✿
∂U 0 0
(x1 , x2 , . . . , x0n , z 0 ) = 0
∂z

▲➜② ✈✐ ♣❤➙♥ ❤➺ t❤ù❝ tr➯♥ t❤❡♦ xi ✈ỵ✐ z ❧➔ ❤➔♠ ❝õ❛ x1, x2, . . . , xn t❛ ✤÷đ❝
∂U
∂U ∂z
+
·
= 0✱
∂xi
∂z ∂xi

∂z
❤❛② ∂x

i

=−

∂U ∂U
:
∂xi ∂z

❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ (1.3) t❛ ✤÷đ❝
✭✶✳✹✮
✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ✤è✐ ✈ỵ✐ ❤➔♠ sè ❯ ❝➛♥ t➻♠✳

X1

∂U
∂U
∂U
+ . . . + Xn
+f
=0
∂x1
∂xn
∂z

●✐↔ sû ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ t÷ì♥❣ ù♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥
dx1
dx2
dxn
dz
=
= ... =
=
X1
X2
Xn
f

❝â ♥ t➼❝❤ ♣❤➙♥ ✤ë❝ ❧➟♣
✶✵


ϕ1 (x1 , x2 , . . . , xn , z), ϕ2 (x1 , x2 , . . . , xn , z), . . . , ϕn (x1 , x2 , . . . , xn , z)


❑❤✐ ➜② ❤➔♠ sè U = Φ(ϕ1, . . . , ϕn) ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❤➺
♣❤÷ì♥❣ tr➻♥❤ (1.4)✱ ❞♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1.3) ❝â
❞↕♥❣
U = Φ(ϕ1 , . . . , ϕn ) = 0
u
u
+y +z
= xyz t ữ ỵ r
ử ố ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ x ∂u
∂x
∂y
∂z
dy
dz
du
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ dx
=
=
=
❝â ❜❛ t➼❝❤ ♣❤➙♥ ✤ë❝ ❧➟♣
x
y
z
xyz
❧➔ ϕ1 = xy , ϕ2 = xz ✱ ϕ3 = xyz − 3u✳ ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ tê♥❣ q✉→t
❧➔
y z
Φ( , , xyz − 3u) = 0
x x


✶✳✹ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët
✣à♥❤ ♥❣❤➽❛ ✹✳ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣
dy
+ P (x)y = Q(x),
dx

✭✶✳✺✮

✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët✳
❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ (1.5) t❛ ❧✉ỉ♥ ♠➦❝ ✤à♥❤ P (x) ✈➔ Q(x) ❧➔ ①→❝ ✤à♥❤
tr➯♥ ❦❤♦↔♥❣ (a, b) ♥➔♦ õ t t ừ (1.5) ữợ
y = u(x)v(x),
✭✶✳✻✮
ð ✤➙② u, v ❧➔ ❝→❝ ❤➔♠ ❝❤÷❛ ❜✐➳t ✈➔ ♣❤ö t❤✉ë❝ ✈➔♦ x✳ ✣↕♦ ❤➔♠ ❝↔ ❤❛✐ ✈➳
❝õ❛ (1.6) t❛ ♥❤➟♥ ✤÷đ❝✿
y = u v + uv ✱
✈➔ ✤➦t ❜✐➸✉ t❤ù❝ ✈ø❛ ♥❤➟♥ ✤÷đ❝ ✈➔♦ (1.5) t❛ ♥❤➟♥ ✤÷đ❝✿
u v + uv + P (x)uv = Q(x)
⇐⇒ uv + (u + P (x)u)v = Q(x)
✶✶

✭✶✳✼✮


❚r♦♥❣ (1.7) t❛ ❝❤å♥ u s❛♦ ❝❤♦
u + P (x)u = 0
⇐⇒ u = −P (x)u
⇐⇒


du
= −P (x)u
dx

⇐⇒

du
= −P (x)dx
u

⇐⇒ ln(u) = − P (x)dx

❚ø ✤➙② t❛ ♥❤➟♥ ✤÷đ❝✿
u = e−

P (x)dx

.

❚❤❛② ❜✐➸✉ t❤ù❝ ♥❤➟♥ ✤÷đ❝ ✈➔♦ (1.7) t❛ ♥❤➟♥ ✤÷đ❝✿
e−

P (x)dx

⇐⇒ e

· v = Q(x)
· e−

P (x)dx


·v =e

dv
=e
dx

P (x)dx

· Q(x)

P (x)dx

⇐⇒ v =

=⇒ dv = e
=⇒ v =

e

P (x)dx

P (x)dx

· Q(x)

· Q(x)dx

P (x)dx


· Q(x)dx + C

❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ qt ừ ữỡ tr (1.5) t ữủ ữợ
y = uv = e−

P (x)dx

·[

e

P (x)dx

❱➼ ❞ö ✻✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✿
xy − 2y = 2x4
✶✷

· Q(x)dx + C].

✭✶✳✽✮


P❤÷ì♥❣ tr➻♥❤ (1.8) ⇐⇒ y
−2
=⇒
P (x) =
x3
Q(x) = 2x

2

− y = 2x3
x
P (x)dx = −2ln(x)

=⇒ y = e2ln(x) · [ e−2ln(x) · 2x3 dx + C]
1
⇐⇒ y = x2 · [ 2 · 2x3 dx + C]
x
⇐⇒ y = x2 [ 2xdx + C]
⇐⇒ y = x2 [x2 + C]

✶✳✺ ữớ ỗ t
ữớ R2 ữủ ồ ữớ ỗ t

t ừ ữớ tỗ t ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐
❆❢❢✐♥❡ ❝❤✉②➸♥ ✤✐➸♠ ♥➔② s❛♥❣ ✤✐➸♠ ❦✐❛ ũ ợ ừ õ
ỳ ữớ ỗ t ❆❢❢✐♥❡ ❧➔ ♥❤ú♥❣ ✤÷í♥❣ ❝♦♥❣ ✤÷đ❝ ❜↔♦
t♦➔♥ ❜➡♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ t❤➼❝❤ ❤ñ♣✳
P❤➨♣ ❜✐➳♥ ✤ê✐ tå❛ ✤ë s❛✉ ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✿
∗

∗

x = a1 x + b 1 y + c 1
, ✈ỵ✐
y = a2 x∗ + b2 y ∗ + c2

❚r♦♥❣ ✤â ✿ ✭①✱②✮ ✲ tồ ở ố

ữ ỵ




a b
1 1
=
=0
a2 b 2 

(x∗ , y ∗ )

✭✶✳✾✮

✲ tå❛ ✤ë ❝❤✉②➸♥ ✤ê✐

P❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✭✶✳✾✮ ❝ị♥❣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ∆ ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ ✈✐➳t✿
x∗ = a1 x + b1 y + c1
y ∗ = a2 x + b2 y + c2

t ổ t ữớ ỗ t tr t
ữ ỵ ữớ ỗ ♥❤➜t ❆❢❢✐♥❡ tr♦♥❣ ♠➦t ♣❤➥♥❣ ❧➔ ❝→❝ ✤÷í♥❣

❝♦♥❣ ❝â t❤➸ ❜✐➳♥ ✤ê✐ t❤➔♥❤ ❝❤➼♥❤ ♥â ❜➡♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡✳
❱➼ ❞ư ✼✳ ✣÷í♥❣ ❝♦♥❣ S 1 = x2 + y2 = 1 ữớ ỗ t



❚❤➟t ✈➟②✱ ♣❤➨♣ q✉❛② ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠ët ❣â❝ ϕ ✿
x∗ = xcosϕ − ysinϕ
y ∗ = xsinϕ + ycosϕ


❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✈➔ ♥â ❜↔♦ t♦➔♥ ✤÷í♥❣ trá♥✳
❈❤å♥ ❣â❝ ϕ = ϕB − ϕA t❤➻ t❛ s➩ t❤✉ ✤÷đ❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➛♥ t➻♠ ♥❤÷
tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✶✳✾✳
✣à♥❤ ❧➼ ✷✳ ❚➜t ❝↔ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ❝➜♣ ❤❛✐✱ ♥❤÷ ❧➔ ✿ ❡❧❧✐♣✱
❤②♣❡r❜♦❧✐❝✱ ♣❛r❛❜♦❧✐❝ ✤➲✉ ❧➔ ữớ ỗ t

ự
t ữớ ✿

x2 y 2
+
=1
a2 b2

✣÷đ❝ ❜✐➳t✱ ✤÷í♥❣ ❡❧❧✐♣ ❧➔ ✤÷í♥❣ trá♥ ❜à ✧♥➨♥✧✳
✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ r➡♥❣ ✈✐➺❝ ♥➨♥ ❞å❝ t❤❡♦ trư❝ tå❛ ✤ë ❝â ♣❤÷ì♥❣ tr➻♥❤✿
x∗ = cx
y∗ = y

❚❤❛② ✭✶✳✶✵✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✤÷í♥❣ trá♥ x2 + y2
✤÷đ❝✿

✭✶✳✶✵✮
= R2 ✱

t❛ ♥❤➟♥

(x∗ )2
+ (y ∗ )2 = R2

2
c
(x∗ )2
(y )2
+
=1
(cR)2
R2

ữỡ tr

ỗ t ờ ✭t✉②➳♥ t➼♥❤✮ ❜✐➳♥ ✤÷í♥❣ ❡❧❧✐♣ t❤➔♥❤ ✤÷í♥❣
trá♥✳
✣✐➸♠ ❆ ✈➔ ❇ s➩ ✤✐ ✤➳♥ ✤✐➸♠ A∗ ✈➔ B ∗ t÷ì♥❣ ù♥❣✳ ❚r♦♥❣ ✤â ❝→❝ ✤✐➸♠
♥➡♠ tr➯♥ ❝ị♥❣ ♠ët ✤÷í♥❣ trá♥ ❝ơ♥❣ tở ữớ trỏ
ữớ trỏ ữớ ỗ t tỗ t ởt
ờ A∗ ❞✐ ❝❤✉②➸♥ ✤➳♥ ✤✐➸♠ A∗∗ = B ∗✳ ❱➔ ♥❤ú♥❣ ✤✐➸♠
✶✹


ợ tr ữớ trỏ tở ✤÷í♥❣ trá♥✳
❱✐➺❝ ❝❤✉②➸♥ ✤ê✐ ♥❣÷đ❝ ❜✐➳♥ ✤÷í♥❣ trá♥ t❤➔♥❤ ❡❧❧✐♣ tr♦♥❣ ✤â ✤✐➸♠ A∗∗ ❞✐
❝❤✉②➸♥ ✤➳♥ ✤✐➸♠ A∗∗∗ = B
❙ü ❦➳t ❤ñ♣ ❝õ❛ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ❜✐➳♥ ✤✐➸♠ t

ữ ỵ ỳ ữớ ự ữớ ỗ t

t tỗ t ♠ët s♦♥❣ →♥❤ ❆❢❢✐♥❡ t❤➻ ✤÷í♥❣ ❝♦♥❣ ✤â ❝ơ♥❣ ❧➔ ữớ
ỗ t


t ữớ r
sỷ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✿

x2 y 2
−
=1
a2 b2

x = x∗ a
y = y∗b

✣➸ ❜✐➳♥ ❤②♣❡r❜♦❧✐❝ t❤➔♥❤✿
(x∗ a)2 (y ∗ b)2
−
= 1 ⇔ x2 − y 2 = 1
2
2
a
b

✭✶✳✶✶✮

❈❤ó♥❣ t❛ sû ❞ư♥❣ ❝→❝ ❤➔♠ ❝õ❛ ❤②♣❡r❜♦❧✐❝ ✭❝❤t ✈➔ s❤t✮
◆❤÷ t❛ ✤➣ ❜✐➳t✱ ❝→❝ ❤➔♠ sinϕ✱ cosϕ ❧➔ ❝→❝ ❤➔♠ ❧÷đ♥❣ ❣✐→❝ ❝ì ❜↔♥ t❤ä❛
♠➣♥✿
sin2 ϕ + cos2 ϕ = 1

❈→❝ ❤➔♠ ❤②♣❡r❜♦❧✐❝ ✿ ch2t − sh2t = 1
❳➨t ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✭t✉②➳♥ t➼♥❤✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ q✉❛② ❤②♣❡r✲
❜♦❧✐❝✿

x∗
x
=
A
✭✶✳✶✷✮
y∗
y
❤♦➦❝
∗
x
−1 x
=A
✭✶✳✶✸✮
y
y∗
✈ỵ✐ A =

cht sht
sht cht

, A−1 =

cht −sht
−sht cht
✶✺

=

ch(−t) sh(−t)
sh(−t) ch(−t)



P❤➨♣ ❜✐➳♥ ✤ê✐ ♥➔② ❜✐➳♥ ✤ê✐ ❤②♣❡r❜♦❧✐❝ ✭✶✳✶✶✮ t❤➔♥❤ ❝❤➼♥❤ ♥â ✈➻ ♥â
t❤ü❝ ❤✐➺♥ ✈✐➺❝ ❜✐➳♥ ✤ê✐ ❝→❝ ✤✐➸♠ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ♥➔② ✈➔ ↔♥❤ ❝õ❛ ❝❤ó♥❣✿
x
y

= A−1

x∗
y∗

=

x∗ ch(−t) + y ∗ sh(−t)
x∗ sh(−t) + y ∗ ch(−t)

x2 − y 2 = (x∗ )2 ch2 (−t) + (y ∗ )2 sh2 (−t) + 2x∗ y ∗ ch(−t)sh(−t)
− (x∗ )2 sh2 (−t) − (y ∗ )2 ch2 (−t) − 2x∗ y ∗ ch(−t)sh(−t) = 1
⇔ (x∗ )2 − (y ∗ )2 = 1

✣✐➸♠

x∗
y∗

∈ Γ✱

❝❤✉②➸♥ t❤➔♥❤ ✤✐➸♠


x
y

✱ ❝ô♥❣ ♥➡♠ tr➯♥ ❤②♣❡r❜♦❧✐❝✳

❈❤➾ r❛ r➡♥❣✱ ❜➡♥❣ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✭t✉②➳♥ t➼♥❤✮✱ ❜➜t ❦➻ ✤✐➸♠
♥➔♦ tr➯♥ ❤②♣❡r❜♦❧✐❝ ❝ơ♥❣ ❝â t❤➸ ✤÷đ❝ ❞à❝❤ s❛♥❣ ❜➜t ❦➻ ✤✐➸♠ ♥➔♦ ❦❤→❝
tr➯♥ ✤÷í♥❣ ❝♦♥❣ ✤â✳
✣➸ ❧➔♠ ✤÷đ❝ ✤✐➲✉ ♥➔②✱ t❛ ❝❤♦ ✤✐➸♠ ▼✭✶❀✵✮ t❤✉ë❝ ♠ët ❤②♣❡r❜♦❧✐❝ ❣✐→❝
✤➲✉✳ ❈❤♦ N (x0; y0) ✭✈ỵ✐ x0 = cht✱ y0 = sht✮✱ ❧➔ ✤✐➸♠ ❝❤✉②➸♥ ✤ê✐ ❝õ❛ ▼
q✉❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡✳
x∗
y∗

=A

1
0

=

N (cht; sht) =

cht
sht

et + e−t et − e−t
,
2
2


❚↕✐ t = arcshy0 ✭❤♦➦❝ t = arcchx0✮ ❧➔ ❝→❝ ✤✐➸♠ ❧➛♥ ❧÷đt ✤÷đ❝ t➻♠✳
❚r➯♥ ✤÷í♥❣ ❝♦♥❣ ❧➜② t❤➯♠ ♠ët ✤✐➸♠ K(x1, y1)✳ ❉ò♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✭✶✳✶✸✮
❝â t❤➸ ❜✐➳♥ K → M
N →K
ϕn
:M →N
(ϕn )−1 : N → M
ϕk
:M →K
✶✻


❑➳t ❤ñ♣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ϕm = ϕk (ϕn)−1 : N → K ❜✐➳♥ ❤②♣❡r❜♦❧✐❝
t❤➔♥❤ ❤②♣❡r❜♦❧✐❝✳ ❱➻ ✈➟②✱ ✤÷í♥❣ ỗ t

t ữớ r y = x2
t ờ tũ ỵ

x = a1 x∗ + b1 y ∗ + c1
y = a2 x∗ + b2 y ∗ + c2

❱➔ ❝❤å♥ ❝→❝ ❤➺ sè ✤➸ ♣❛r❛❜♦❧✐❝ ❜✐➳♥ ✤æ➾ t❤➔♥❤ ❝❤➼♥❤ ♥â✳
a2 x∗ + b2 y ∗ + c2 = (a1 )2 (x∗ )2 + (b1 )2 (y ∗ )2 + (c1 )2 + 2a1 b1 x∗ y ∗ + 2a1 c1 x∗ +
2b1 c1 y ∗

❈❤å♥ a1 = 1, b1 = 0, b2 = 1✳ ❑❤✐ ✤â ✿
a2 x∗ + y ∗ + c2 = (x∗ )2 + c21 + 2c1 x∗ ✱ s✉② r❛ y ∗ ✳
y ∗ = (x∗ )2 +2c1 x∗ +c21 −c2 −a2 x∗ ❤♦➦❝ y ∗ = (x∗ )2 +2(c1 −a2 )x∗ +(c21 −c2 )
❚❛ ❝â✿

a2 = 2c1
2c2 − a2 = 0
,
❤♦➦❝
2
2
c1 − c2 = 0

❑❤✐ ✤â✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ trð t❤➔♥❤ ✿

c2 = c1

x = x∗ + c
y = 2cx∗ + y ∗ + c2

✭✶✳✶✹✮

♣❛r❛❜♦❧✐❝ t ợ tũ ỵ
t tr r ●✐↔ sû N (x0; y0) ❧➔ ♠ët ✤✐➸♠ ❝ô♥❣ ♥➡♠
tr➯♥ ♣❛r❛❜♦❧✐❝✳ ❚❛ sû ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❆❢❢✐♥❡ ✭✶✳✶✹✮ ♠ët ❝→❝❤ ♣❤ị ❤đ♣
✤➸ ❜✐➳♥ ✤✐➸♠ ▼ t❤➔♥❤ ✤✐➸♠ N (x0; y0)✳
✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ r➡♥❣✱ ❝➛♥ ♣❤↔✐ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿
1 = x0 + c
1 = 2cx0 + y0 + c2

✭✶✳✶✺✮

❚ø ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ t❛ ❝â c = 1 − x0✱ ❦❤✐ ✤â t❛ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ♣❤÷ì♥❣
tr➻♥❤ tự ừ ữỡ tr
ữỡ tỹ ợ ❧➟♣ ❧✉➟♥ ✤÷đ❝ →♣ ❞ư♥❣ ❝❤♦ ✤÷í♥❣ ❤②♣❡r❜♦❧✐❝ t❛ ❝❤ù♥❣

♠✐♥❤ ữủ r ữớ ỗ t



ữỡ
ì
P ể
ử ✈➲ ✤↕✐ sè ♠❛ tr➟♥
✣è✐ t÷đ♥❣ ❝❤➼♥❤ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ♠ët ✤↕✐ sè tr÷í♥❣
✈❡❝t♦r t✉②➳♥ t➼♥❤ t❤ü❝ ✺ ❝❤✐➲✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♣❤ù❝ C3✳ ❚å❛ ✤ë tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ ♥➔② s➩ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ z1✱ z2✱ u✳
▼é✐ t❤➔♥❤ ♣❤➛♥ ❝õ❛ ✤↕✐ sè t❤↔♦ ❧✉➟♥ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr÷í♥❣
✈❡❝t♦r ❝â ❞↕♥❣✿
E = (A1 z1 + A2 z2 + p)
(az1 + bz2 + cw)

∂
∂w

∂
∂
+ (B1 z1 + B2 z2 + s)
+
∂z1
∂z2
(1)

❚❤❡♦ tr÷í♥❣ ✈❡❝t♦r ❝â t❤➸ ❧➜② ✈✐ ♣❤➙♥ ❤➔♠✱ ♠➔ ❤➔♠ ♥➔② ✤÷đ❝ ♥❤➜♥
♠↕♥❤ ❜➡♥❣ ✈✐➺❝ sỷ ử (1) ữợ ú t s ❞ò♥❣ ❝→❝ ♣❤➨♣
t♦→♥ ✈✐ ♣❤➙♥ ✭♣❤➨♣ t➻♠ ✤↕♦ ❤➔♠✮ t❤❡♦ tr÷í♥❣ ✈❡❝t♦r✳ ◆❤÷♥❣ ✤➸ ✈✐➺❝ ♠ỉ

t↔ ✤÷đ❝ rã r➔♥❣ ❤ì♥✱ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ❝→❝ tr÷í♥❣ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣
♠❛ tr➟♥✳
✣➦❝ ❜✐➺t✱ ✺ ♠❛ tr➟♥ s❛✉ ✤➙② ❧➔ ❝ì sð ❝õ❛ ✤↕✐ sè ♠❛ tr➟♥ ♠➔ ❝❤ó♥❣ t❛
s➩ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥✿
✶✽





15
−57
1 0
0
0
 0
 1

128
16 













15 
1
−9 


 0
 0

✵ 32 
0 0



2
8 
 , E2 = 
,
E1 = 




 21i 21i
 −21i −147i 3







0 
0 0 


 8
 16


128 2
8








0
0
0 0
0
0 0 0



15i
i 0

 0
16 







0 0 0
−i 

 0 0 0




8 
 , E4 =  0 0 0
E3 = 


 0 −3 0
 3 −21




0 0 


0 0 0
 8 8





0 0 0 0


❇ð✐




Ek = 



i
0
0
0










 , E5 = 





0
0
0
0

0
0
0
0

0
0
0
0

0
0
1
0


✭k)

✭k)
A1 A2
0 pk

✭k)
✭k)
B1 B2
0 sk 
,
ak
b k ck 0 

0
0
0 0

Ð ✤➙② Ek ❧➔ ❦➼ ❤✐➺✉ ♠❛ tr➟♥ ♠➔ ❝→❝ ♠❛ tr➟♥ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ tr÷í♥❣
✈❡❝t♦r t✉②➳♥ t➼♥❤ ❝ì sð✳ ❇✐➳t r➡♥❣ tr÷í♥❣ ♥➔② t❤✉ë❝ tr♦♥❣ ✤↕✐ sè ♠➔
❝❤ó♥❣ t❛ ✤❛♥❣ ♥❣❤✐➯♥ ❝ù✉✿
✭k)

✭k)

Ek = (A1 z1 + A2 z2 + pk )
(ak z1 + bk z2 + ck w)

∂
✳
∂w


∂
∂
✭k)
✭k)
+ (B1 z1 + B2 z2 + sk )
+
∂z1
∂z2

✶✾




.




ữ ỵ r tr q tr ờ tứ tr÷í♥❣ ✈❡❝t♦r t❤➔♥❤ ♠❛
tr➟♥✱ ❝→❝ ♣❤➨♣ t♦→♥ ♥❣♦➦❝ ✈✉ỉ♥❣ ❝õ❛ ❤❛✐ tr÷í♥❣ ✈❡❝t♦r s➩ ❝❤✉②➸♥ t❤➔♥❤
♣❤➨♣ t♦→♥ ♥❣♦➦❝ ✈✉ỉ♥❣ ❝õ❛ ❝→❝ ♠❛ tr➟♥ t÷ì♥❣ ù♥❣✱ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣
t❤ù❝ s❛✉✿
[A, B] = A · B − B · A

❚❤➟t ✈➟②✿
•

[E1 , E2 ] = E1 · E2 − E2 · E1




1 0
 0




 0
0 0

=⇒ [E1 , E2 ] = 

 63i 21i

0

 16 8


0
0 0

 
15
16 
 
 
 
−9  

 
16  
−
 

63i 
 
 
1024  
 
 
0

0

1
2

0

0

0

0

21i 21i
0
8 16
0


0

0




15
15
0
1 0
 0
 0
32 
16 












−9 
−9 



 0

 0

0 0
0 0

16  1 
8  1
= ·
 = E2
=

 2 
 2
 21i 21i

 21i 21i





0 0 
0 0 


 16 16


 8

8








0
0 0 0
0
0 0 0


•

1
2

[E1 , E3 ] = E1 · E3 − E3 · E1

✷✵


15
32 




0 



63i 


1024 


0




i 0
 0




 0
0 0

=⇒ [E1 , E3 ] = 

 9 −21


0

 16 8


0
0 0

 
15i
16 
 
 
 
−i  
 

16 
−
 

1113 
 
 
1024  
 
 
0


1
i
2

0

1
i
2

0

1
159
= E3 +
E5
2
64

•

[E1 , E4 ] = E1 · E4 − E4 · E1

✷✶









0 0 0
0



−1431 
3 −21

0

8 16
1024 


0 0 0
0




15
15i

0
i 0
0
i 
 0



32 
16 













−1 
−i 



 0
 0 0 0

0 0
i

 1
 159 

16
8

 = ·
+
=

 2
 64 ·

 3 −21

 3 −21


159





0
0 0 




 16 16
 8 8


64 









0
0 0 0
0 0 0 0


15
i
32

0 0 0 0






0 0 0 0





0 0 0 1



0 0 0 0




0

0

0

i





0

0

0

0




 


 


 


0 0 0 0  0 0 0 0 
 


 


−

=⇒ [E1 , E4 ] = 
 


−3
−9
21  
−45 

 0


0
0
0
 


2
16
2
32
 


 


0 0 0 0
0 0 0 0


0

0

0





0 0 0


=


 0 −3 0



0 0 0
87
= E3 +
· E5
32

•

i





0

0

0 i






0 0 0 0



 



 



 




0 0 0 0
0  0 0 0 0

 87 
 



=

+
·





 32 

87  

0 0 0 1
0
−3
0
0
 




32 



 



 

0 0 0 0
0 0 0 0
0

[E1 , E5 ] = E1 · E5 − E5 · E1



0 0 0 0





0 0 0 0
















0 0 0 0 


0 0 0 0

 3
 3 





=⇒ [E1 , E5 ] = 
 = · E5
= 2 ·

 2
3 

0 0 0 1

0 0 0



2 









0 0 0 0
0 0 0 0
✷✷


•

[E2 , E3 ] = E2 · E3 − E3 · E2



0 0 0




0 0 0


=⇒ [E2 , E3 ] = 

−21

0
0


8


0 0 0


0

0

0

i





0

 
−i

8 
 
 
 

0 
 

 
−
 
−273  
 
128  
 
 
0

0

0 i



0 0
0 0
0

3
8

0 0



−9i
0
8 




0 0 




423 
0

128 


0 0

0 0 0 0




 




 





 



0 0 0 0  




 0 0 0 0
0 0 0 0

 
87



=
=
·
−


 
 16 

−87  

0 0 0 1

 0 −3 0
  0 −3 0 0 




16 




 




 
0 0 0 0
0 0 0 0
0 0 0 0

= E4 −

•

87
· E5
16

[E2 , E4 ] = E2 · E4 − E4 · E2


✷✸




0 0 0

0





0 0 0

0



 


 


 


0 0 0 0  0 0 0 0 

 


 


−

=⇒ [E2 , E4 ] = 
 


27 
−21  

 0 0 0

0 0 0
 


8
8
 


 


0 0 0 0

0 0 0 0


0 0 0

0





0 0 0 0















0 0 0 0
0 0 0 0 









=
 = −6 · E5
 = −6 · 




0 0 0 1
 0 0 0 −6 












0 0 0 0
0 0 0 0


•

[E2 , E5 ] = E2 · E5 − E5 · E2

=⇒ [E2 , E5 ] = 0
•

[E3 , E4 ] = E3 · E4 − E4 · E3


0 0 0 0





0 0 0 0


 

 

 
0 0 0 0  0 0 0

 

 

−
=⇒ [E3 , E4 ] = 

 
3i  

0 0 0
 0 0 0

 
8

 

 
0 0 0 0
0 0 0

✷✹






0 


=0


3i 

8 


0