❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✣⑨ ◆➂◆● ❑❍❖❆ ❚❖⑩◆ ◆●❯❨➍◆ ❚❍➚ P❍×Đ◆● ✣❸■ ❙➮ ❚❘×❮◆● ❱❊❈❚❖❘ ❱⑨ ❚➑❈❍ P❍❹◆ ❈❍Ĩ◆● ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ◆❣➔♥❤✿ ❈û ♥❤➙♥ ❚♦→♥✲❚✐♥ ●✐↔♥❣ ✈✐➯♥ ữợ ề ì ▼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➼♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữớ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ✣❸■ ❙➮ ❚❘×❮◆● ❱❊❈❚❖❘ ❱⑨ ❚➑❈❍ P❍❹◆ ❈❍Ó◆● ✶✽ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ❱➼ ❞ö ✈➲ ✤↕✐ 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♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ữợ sỹ ữợ t t ❝ỉ ❣✐→♦ ✲ ◆❣✉②➵♥ ❚❤à ❚❤ị② ❉÷ì♥❣✳ ❊♠ ①✐♥ ❣û✐ ✤➳♥ ❝ỉ ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❦❤â❛ ❧✉➟♥✱ t❤æ♥❣ q✉❛ ❝→❝ ❜➔✐ ❣✐↔♥❣✱ ❜➔✐ ❤å❝✱ ❡♠ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sỹ q t ú ù ỳ ỵ õ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦✱ ❚❙✱❚❤❙ t❤✉ë❝ ❦❤♦❛ ❚♦→♥ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✳ ❚ø ✤→② ❧á♥❣ ♠➻♥❤✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝ t❤➛② ❝æ✳ ❊♠ 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➻♥❤ ➞♥✮✳ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ ỵ ữỡ tr ú t õ t❤➸ ❝✉♥❣ ❝➜♣ ♥❤ú♥❣ ❝→❝❤ ❦❤→❝✳ ❱➼ ❞ö✱ ❝â ♠ët trữớ ữợ tữỡ ữỡ õ ởt ữớ ❝♦♥❣✮ 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 trũ ợ ữợ 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 trữợ tr ợ ỵ t sỹ tỗ t t t ữỡ tr ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝❤ó♥❣ t❛ ♣❤↔✐ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ✈➔ ❞♦ ✤â✱ ♣❤↔✐ ✤è✐ õ ợ ỵ ự t ỡ 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 sè ✤â✳ ❚r♦♥❣ ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔② ❣✐↔✐ q✉②➳t ❜➔✐ t q ự t ỗ ♥❤➜t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ✸ ❝❤✐➲✉✳ ✸ ❈ư t❤➸ ❧➔✱ ❝❤♦ ♠ët ✤↕✐ sè tr÷í♥❣ ✈❡❝t♦r t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ C3 ✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ữủ s tỹ ỗ t s q✉❛ ❣è❝ tå❛ ✤ë ✈➔ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ♥â õ sỹ t ú ợ trữớ t sè ✤➣ ❝❤♦✳ ▼ët ❝→❝❤ tü ♥❤✐➯♥✱ ❝â t❤➸ ♥â✐ r ú trữớ t ú ợ t ♠➔ ❝❤ó♥❣ t❛ ✤❛♥❣ t❤↔♦ ❧✉➟♥✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ①➙② ❞ü♥❣ ❜➲ ♠➦t ❞ü❛ tr➯♥ ✤↕✐ sè ❝→❝ tr÷í♥❣ t ú ợ t t ỹ ữủ tr õ ởt tr ỳ s ỗ ♥❤➜t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ C3 ✳ ✣➔ ◆➤♥❣✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à P❤÷đ♥❣ ✹ ❈❤÷ì♥❣ ✶ ◆❍Ú◆● ❑❍⑩■ ◆■➏▼ ❈❒ ❇❷◆ ✶✳✶ ❇❛♦ t✉②➳♥ t➼♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳ ●✐↔ sû ❆ ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ổ tr ổ ổ tỗ t ởt ổ ❣✐❛♥ ❝♦♥ ❝õ❛ ❳ ❝❤ù❛ ❆✳ ●✐❛♦ ❝õ❛ ❤å t➜t ❝↔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝❤ù❛ ❆ ❝ơ♥❣ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝❤ù❛ ❆✳ ❑❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ 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 = 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] ✤÷đ❝ ❣å✐ ❧➔ t➼❝❤ ▲✐❡ ❝õ❛ ① ✈ỵ✐ ② ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ s❛✉✿ ✭✐✮ (L1) : [., ] s♦♥❣ t✉②➳♥ t➼♥❤✳ ✭✐✐✮(L2) : [., ] ♣❤↔♥ ①ù♥❣ ✿ [x, x] = ∀x ∈ (L3) : [., ] tọ ỗ t [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = ❱➼ ❞ö ✶✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ R3 ợ t t õ ữợ sỡ ❝➜♣ ❧➔ ♠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) : [., ] tọ 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✳ R3 ợ t ữ tr ởt sè ▲✐❡✳ ◆❤➟♥ ①➨t ✶✳ • ❚r➯♥ ♠é✐ K ❜➜t ❦➻✱ ▲ ✤➲✉ ❝â t❤➸ tr❛♥❣ ❜à t➼❝❤ ▲✐❡ t➛♠ t❤÷í♥❣ [x, y] = ∀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 ❝❤✐➲✉ ❝õ❛ ✤↕✐ số số ã Pữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ❈â ♥❤✐➲✉ ❜➔✐ t♦→♥ tỹ t t ỵ tt tỵ✐ ✈✐➺❝ t➻♠ ♠ët ❤➔♠ z(x1, x2, , xn)✱ ❜✐➳♥ x1, x2, , xn✱ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ∂z ∂z ∂ 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) ✈➔ ❦❤æ♥❣ ❝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 + + 5u =0 ∂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) ✼ ◆➳✉ ✈➳ ♣❤↔✐ ❝õ❛ ữỡ tr tr ỗ t ỏ 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 ) = ∂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 + = ❝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 ) = ∂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➡♥❣ ♥â ❝â ✈ỉ sè ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✳ ũ ợ ữỡ tr r t 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 ≡ ∂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 ✤è✐ 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 = t ú ỵ r x y z dy dz = = ✳ ❍➺ ♥➔② ❝â ❤➺ ♣❤÷ì♥❣ 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 )✳ ✾ 15 ∂F ∂F 21 21 ∂F ) + y2 − − x1 − x2 = 16 ∂x1 ∂y1 ∂x2 8 ⇐⇒ (x2 + ⇐⇒ (16x2 +15) •Re (z1 − ∂F ∂F ∂F −18 +16y2 −42x1 −42x2 = ∂x1 ∂x2 ∂y1 57 ∂F 15 ∂F 21 147 ∂Φ ) +( z2 + ) +(− iz1 − iz2 + w) 128 ∂z1 32 ∂z2 16 128 ∂w (3 ) ≡0 57 ∂F ∂F 15 ∂F ) ( −i )+( (x2 +iy2 )+ )· ( − 128 ∂x1 ∂y1 32 ∂x2 ∂F 21 147 i i ) + (− i(x1 + iy1 ) − i(x2 + iy2 ) + (u + iF )) ≡0 ∂y2 16 128 2 ⇐⇒ Re ((x1 +iy1 )− ∂F ∂F 21 15 ∂F 57 ∂F ) + ( x2 + ) + y1 + y2 + x1 + 128 ∂x1 32 ∂x2 ∂y1 ∂y2 16 ∂F 57 ∂F 147 ∂F 15 ∂F + y2 − (x1 − − ( x2 + ) + x2 − F ) + i(y1 ) 128 ∂x1 ∂x2 128 ∂y1 32 ∂y2 21 147 y1 + y2 + u) ≡ 16 128 ⇐⇒ Re ((x1 − ⇐⇒ (x1 − ∂F 57 ∂F 15 ∂F ∂F 21 ) + ( x2 + ) + y1 + y2 + x1 + 128 ∂x1 32 ∂x2 ∂y1 ∂y2 16 147 x2 − F = 128 ∂F ∂F ∂F ∂F + (64x2 + 60) + 128y1 + 64y2 + ∂x1 ∂x2 ∂y1 ∂y2 168x1 + 147x2 − 192F = (4 ) ⇐⇒ (128x1 − 57) ❚ø (1 )✱ (2 )✱(3 )✱ (4 )✱ t❛ s➩ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4) ð tå❛ ✤ë t❤ü❝ x1, x2, y1, y2✱ u, v tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ C3 ỗ ữỡ tr s F F ∂F ∂F (128x − 57) + (64x + 60) + 128y + 64y + 168x1 + 2 ∂x ∂x ∂y ∂y 2 147x2 − 192F = ∂F ∂F ∂F (16x2 + 15) − 18 + 16y2 − 42x1 − 42x2 = ∂x1 ∂x2 ∂y1 ∂F ∂F ∂F (16x + 15) − 16y − − 6y1 + 42y2 = (4) 2 ∂y1 ∂x1 ∂y2 ∂F + 3y2 = ∂y1 ✷✳✸ P❤➨♣ ❧➜② t➼❝❤ ♣❤➙♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❈❤ó♥❣ t❛ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4) tr➯♥ t❤❡♦ t❤ù tü✱ tø♥❣ ữợ số ữủ ữỡ tr số ữủ tr sỡ ỗ t ♥❤÷ ✈➟② ✤➣ ✤÷đ❝ t❤ü❝ ❤✐➺♥✱ ✈➼ ❞ư✱ tr♦♥❣ ❦➳t q✉↔ ✭✹✮✮✳ ❈❤ó♥❣ t❛ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤ù (4) ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4)✿ ∂F + 3y2 = ∂y1 ❚ø ⇐⇒ ∂F + 3y2 = ∂y1 ∂F = −3y2 ∂y1 dy1 dx1 dx2 dy2 dF = = = = 0 −3y2 1 x1 = C1 x2 = C2 ⇐⇒ y2 = C dy dF 1= −3y2 ⇐⇒ =⇒ dF = −3y2 dy1 ✸✶ =⇒ F = −3y1 y2 + C4 =⇒ C4 = H(C1 , C2 , C3 ) =⇒ F = −3y1 y2 + H(x1 , x2 , y2 ) ❚ø ✤â t❛ s✉② r❛ ✤÷đ❝✿ ∂F ∂x1 ∂F ∂x2 ∂F ∂y1 ∂F ∂y2 ✈ỵ✐ H ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ ✸ ❜✐➳♥✳ ∂H ∂x1 ∂H = ∂x2 = = −3y2 = −3y1 + ∂H ∂y2 ❚❤❛② ❝→❝ ❣✐→ trà ✈➔♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝á♥ ❧↕✐ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4)✱ t❛ õ ã Pữỡ tr tự t ữỡ tr (4)✿ (128x1 −57) 192F = ∂F ∂F ∂F ∂F +(64x2 +60) +128y1 +64y2 +168x1 +147x2 − ∂x1 ∂x2 ∂y1 ∂y2 ⇐⇒ (128x1 − 57) ∂H ∂H + (64x2 + 60) + 128y1 (−3)y2 + 64y2 (−3y1 + ∂x1 ∂x2 ∂H + 168x1 + 147x2 − 192(−3y1 y2 + H) = ∂y2 ⇐⇒ (128x1 − 57) 192H = • ∂H ∂H ∂H + (64x2 + 60) + 64y2 + 168x1 + 147x2 − ∂x1 ∂x2 ∂y2 P❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4)✿ (16x2 + 15) ∂F ∂F ∂F − 18 + 16y2 − 42x1 − 42x2 = ∂x1 ∂x2 ∂y1 ⇐⇒ (16x2 + 15) ∂H ∂H − 18 + 16y2 (−3y2 ) − 42x1 − 42x2 = ∂x1 ∂x2 ✸✷ ⇐⇒ (16x2 + 15) • ∂H ∂H − 18 − 48y22 − 42x1 − 42x2 = ∂x1 ∂x2 P❤÷ì♥❣ tr➻♥❤ t❤ù ❜❛ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (4)✿ (16x2 + 15) ∂F ∂F ∂F − 16y2 −2 − 6y1 + 42y2 = ∂y1 ∂x1 ∂y2 ⇐⇒ (16x2 + 15)(−3y2 ) − 16y2 ⇐⇒ −16y2 ∂H ∂H − 2(−3y1 + − 6y1 + 42y2 = ∂x1 ∂y2 ∂H ∂H −2 − 48x2 y2 3y2 = x1 y2 ữợ t❛ ❝â ❤➺ ✸ ♣❤÷ì♥❣ tr➻♥❤ ✲ ✸ ➞♥ s❛✉✿ ∂H ∂H ∂H (128x − 57) + (64x + 60) + 64y + 168x1 + 147x2 − 192H = 2 ∂x1 ∂x2 ∂y2 (16x + 15) ∂H − 18 ∂H − 48y − 42x − 42x = 2 ∂x1 ∂x2 ∂H ∂H −16y − − 48x2 y2 − 3y2 = ∂x ∂y ❈❤ó♥❣ t❛ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤ù (3) ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (5)✿ −16y2 ∂H ∂H −2 − 48x2 y2 − 3y2 = ∂x1 ∂y2 ✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ❜➟❝ ♥❤➜t t✉②➳♥ t➼♥❤✳ ✣➸ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ❝❤✉②➸♥ s❛♥❣ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➻♥❤ t❤÷í♥❣✳ −16y2 ∂H ∂H −2 − 48x2 y2 − 3y2 = ∂x1 ∂y2 ⇐⇒ −16y2 ∂H ∂H −2 = 48x2 y2 + 3y2 ∂x1 ∂y2 ✸✸ dx1 dy2 dx2 dH = = = −16y2 −2 48x2 y2 + 3y2 dx1 dy2 = −16y2 −2 dy2 = dx2 −2 ⇐⇒ dy2 dH = −2 48x2 y2 + 3y2 ⇐⇒ ❚ø dx2 dy2 = −2 =⇒ x2 = C5 ❚ø dx1 dy2 = −16y2 −2 =⇒ dx1 = 8y2 dy2 =⇒ x1 = 4y22 + C6 =⇒ C6 = x1 − 4y22 ❚ø dy2 dH = −2 48x2 y2 + 3y2 =⇒ dH = − 48x2 y2 + 3y2 dy2 =⇒ H = −12x2 y22 − y22 + C7 Φ(C5 , C6 , C7 ) = =⇒ C7 = G(C5 , C6 ) ✸✹ =⇒ H = −12x2 y22 − y22 + G(x2 , x1 − 4y22 ) ❜✐➳♥✳ ✈ỵ✐ ● ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ ✷ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ t➼♥❤ t♦→♥✱ ❝❤ó♥❣ t❛ s➩ t❤❛② ✤ê✐ ❝→❝ ❜✐➳♥✳ t1 = x2 t2 = x1 − 4y22 (6) =⇒ x1 = t2 + 4y22 ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ t➼♥❤ ✤÷đ❝✿ ∂G ∂H = ∂x1 ∂t2 ∂H ∂G = − 12y22 ∂x2 ∂t1 ∂G ∂H = −8y2 − 24x2 y2 − y2 ∂y2 ∂t2 ❚❤❛② ❝→❝ ❣✐→ trà tr➯♥ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ♥❤➜t ✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr (5) t õ ã Pữỡ tr tự t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (5)✿ (128x1 −57) ∂H ∂H ∂H +(64x2 +60) +64y2 +168x1 +147x2 −192H = ∂x1 ∂x2 ∂y2 ∂G ∂G ∂G + (64t1 + 60)( − 12y22 ) + 64y2 (−8y2 − ∂t2 ∂t1 ∂t2 3 24x2 y2 − y2 ) + 168(t2 + 4y22 ) + 147t1 − 192(−12x2 y22 − y22 + G) = ⇐⇒ (128(t2 + 4y22 ) − 57) ∂G ∂G + (64t1 + 60) + 168t2 + 147t1 − 192G = ∂t2 ∂t1 tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺ ữỡ tr (5) (128t2 57) ã Pữỡ (16x2 + 15) ∂H ∂H − 18 − 48y22 − 42x1 − 42x2 = ∂x1 ∂x2 ✸✺ ⇐⇒ (16t1 + 15) ∂G ∂G − 18( − 12y22 ) − 48y22 − 42(t2 + 4y22 ) − 42t1 = ∂t2 ∂t1 ⇐⇒ (16t1 + 15) ∂G ∂G − 18 − 42t1 42t2 = t2 t1 ữợ t t❤➳ ❝→❝ ❣✐→ trà ✈➔♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (5)✱ t❛ ❝â ❤➺ ✷ ♣❤÷ì♥❣ tr➻♥❤ ❝á♥ ❧↕✐ t❤❡♦ ❦➳t q✉↔ ❝õ❛ ♣❤➨♣ ❣✐↔✐ ♥❤÷ s❛✉✿ ∂G ∂G (128t − 57) + (64t + 60) + 168t2 + 147t1 − 192G = ∂t2 ∂t1 ∂G ∂G (16t1 + 15) − 18 − 42t1 − 42t2 = ∂t2 ∂t1 (7) ❈❤ó♥❣ t❛ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ✷ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (7) ✿ (16t1 + 15) ∂G ∂G − 18 − 42t1 − 42t2 = ∂t2 t1 ữ trữợ õ õ ữỡ tr t t➼♥❤ ✈➔ ✤➸ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝ơ♥❣ ❝➛♥ ❝❤✉②➸♥ s❛♥❣ ✶ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➻♥❤ t❤÷í♥❣✳ (16t1 + 15) ∂G ∂G − 18 − 42t1 − 42t2 = ∂t2 ∂t1 ⇐⇒ (16t1 + 15) ⇐⇒ ⇐⇒ ∂G ∂G − 18 = 42t1 + 42t2 ∂t2 ∂t1 dt2 dt1 dG = = 16t1 + 15 −18 42t1 + 42t2 dt2 dt1 = 16t1 + 15 −18 dG dt1 = −18 42t1 + 42t2 ✸✻ ❚ø dt2 dt1 = 16t1 + 15 −18 =⇒ (16t1 + 15)dt1 = −18dt2 =⇒ 8t21 + 15t1 = −18t2 + 18C8 =⇒ C8 = t21 + t1 + t2 =⇒ t2 = C8 − t21 − t1 ❚ø dt1 dG = −18 42t1 + 42t2 =⇒ −3dG = (7t1 + 7t2 )dt1 =⇒ dG = − (t1 + C8 − t21 − t1 )dt1 =⇒ dG = − (− t21 + t1 + C8 )dt1 =⇒ G = 7 28 t1 − t21 − C8 t1 + C9 81 36 (∗) ❚❤➳ C8 = 49 t21 + 56 t1 + t2 ✈➔♦ (∗)✱ t❛ ❝â✿ G= = 7 28 t1 − t21 − ( t21 + t1 + t2 ) + C9 81 36 28 56 105 t1 − t21 − t31 − t − t1 t2 + C 81 36 54 44 =− 56 77 t − t − t1 t2 + C9 81 36 Φ(C8 , C9 ) = =⇒ C9 = Q(C8 ) ✸✼ =⇒ G = − 56 77 t1 − t1 − t1 t2 + Q( t21 + t1 + t2 ) 81 36 ▼ët ❧➛♥ ♥ú❛✱ t❛ t❤❛② ✤ê✐ ❝→❝ ❜✐➳♥✿ r = t21 + t1 + t2 (8) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ t➼♥❤ ✤÷đ❝✿ ∂G ∂Q 56 77 = ( t1 + ) − t21 − t1 − t2 ∂t1 ∂r 27 18 (∗∗) ∂G ∂Q = − t1 ∂t2 ∂r ❚❤❛② ❝→❝ ❣✐→ trà ❝õ❛ (∗∗) ✈➔♦ ❜✐➸✉ t❤ù❝ s❛✉✿ (16t1 + 15) ∂G ∂G − 18 − 42t1 − 42t2 ∂t2 ∂t1 = (16t1 +15)( 77 ∂Q ∂Q 56 − t1 )−18( ( t1 + )− t21 − t1 − t2 )−42t1 −42t2 ∂r ∂r 27 18 ∂Q ∂Q ∂Q 112 112 ∂Q − t1 +15 −35t1 + t1 +77t1 +42t2 −16t1 −15 − ∂r ∂r ∂r ∂r 42t1 − 42t2 = 16t1 =0 ❚❤❛② ❝→❝ ❣✐→ trà ❝õ❛ (∗∗) ✈➔ G ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ✶ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ (7)✿ (128t2 − 57) ∂G ∂G + (64t1 + 60) + 168t2 + 147t1 − 192G = ∂t2 ∂t1 ∂Q ∂Q 56 77 ⇐⇒ (128t2 − 57)( − t1 ) + (64t1 + 60)( ( t1 + ) − t21 − t1 − ∂r ∂r 27 18 56 77 t2 ) + 168t2 + 147t1 − 192(− t1 − t1 − t1 t2 + Q) = 81 36 ✸✽ ∂Q 70 320 ∂Q 512 ∂Q ∂Q 112 ⇐⇒ 128t2 + t1 − + t1 + t1 + t + ∂r ∂r 3 ∂r ∂r 28t2 − 192Q = ❚❤➳ t2 = r − 49 t21 − 65 t1 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✱t❛ ✤÷đ❝✿ ∂Q 112 ∂Q 70 320 ∂Q 512 ∂Q 128(r − t21 − t1 ) + t1 − + t1 + t1 + t + ∂r ∂r 3 ∂r ∂r 28(r − t21 − t1 ) − 192Q = ⇐⇒ (128r − 7) ∂Q + 28r − 192Q = ∂r ❤❛② (128r − 7)Q − 192Q = −28r =⇒ Q − 28r 192 Q=− 128r − 128r − P (r) = − 192 128r − K(r) = − 28r 128r − ▲➜② t➼❝❤ ♣❤➙♥ P (r)✳ I= P (r)dr = (− 192 )dr 128r − ✣➦t u = 128r − =⇒ du = 128dr =⇒ dr = du 128 ❉♦ ✤â✱ I = (− 192 du )· =− u 128 du = − ln(u) u ✸✾ K(r) = − 28r 128r − ✣➦t u = 128r − =⇒ K( −28 u + u+7 )= · 128 u 128 =⇒ K( u+7 (u + 7) )=− · 128 32 u =⇒ Q = e− = u2 · [ −3 ln(u) ·[ e −3 ln(u) −1 ( u− · du + 128 32 −1 · (− (u + 7) · ) du + C] 32 u 128 u− · 49 du) + C] 32u −3 49 u −7 u =u ·[ + C] · −1 − · 4096 4096 −3 = 49 ·u+ + C · u2 2048 6144 ❚❤❛② u = 128r − ✈➔♦ Q t❛ ❝â✿ Q= 49 (128r − 7) + + C · (128r − 7) 2048 6144 ⇐⇒ Q = 49 49 r− + + C · (128r − 7) 16 2048 6144 ⇐⇒ Q = 49 r− + C · (128r − 7) 16 3072 • Q= ❚❤❛② r = t2 + 49 t21 + 65 t1 ✈➔♦ Q t❛ ❝â✿ 49 (t2 + t21 + t1 ) − + C · (128(t2 + t21 + t1 ) − 7) 16 3072 ❚✐➳♣ tö❝ t❤❛② t1 = x2, t2 = x1 − 4y22 ✈➔♦ Q✱ t❛ ❝â✿ ✹✵ Q= = 49 (x1 −4y22 + x22 + x2 )− +C ·(128(x1 −4y22 + x22 + x2 )−7) 16 3072 7 35 49 512 320 x1 − y22 + x22 + x2 − +C(128x1 −512y22 + x2 + x2 −7) 16 36 96 3072 • ❚❤❛② t1 = x2, t2 = x1 − 4y22 ✈➔♦ G t❛ ❝â✿ G=− =− 56 77 t − t − t1 t2 + Q 81 36 56 77 x2 − x2 − x2 (x1 − 4y22 ) + Q 81 36 ❙❛✉ ❦❤✐ trð ✈➲ tå❛ ✤ë ❜❛♥ ✤➛✉ ✈ỵ✐ ❝→❝ ♣❤➨♣ t❤❛② t❤➳ ❝❤ó♥❣ t❛ ✤➣ ✤➦t✱ ❝❤ó♥❣ t❛ t❤✉ ✤÷đ❝ ❤➔♠ s❛✉ ✤➙②✿ (6) ✈➔ (8) ♠➔ 56 77 7 V = −3y1 y2 − 12x2 y22 − y22 − x32 − x22 − x2 (x1 − 4y22 ) + x1 − 81 36 16 35 512 320 49 y2 + x + x − + C(128x1 − 512y22 + x22 + x2 − 7) 36 96 3072 35 35 49 + C(128x1 − 512y22 + = −3y1 y2 + x1 − y22 − x22 + x2 − 16 18 96 3072 512 320 x2 + x2 − 7) − x2 (x1 − 4y2 ) − 12x2 y22 3 ❇➲ ♠➦t t❤✉ ✤÷đ❝ ❝❤➼♥❤ ❧➔ ❜➲ ♠➦t ✤↕✐ sè ❜➟❝ ✻✳ ✷✳✹ ❙û ❞ö♥❣ ❝→❝ ❣â✐ t♦→♥ ❤å❝ ▼❛♣❧❡ ✈➔ ▼❛t❤❈❛❞ ❍➛✉ ❤➳t ❝→❝ t➼♥❤ t♦→♥ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜➡♥❣ t❛②✳ ◆❤÷♥❣ ✤➸ t➼♥❤ t♦→♥ ❝â ✤ë t✐♥ ❝➟② ❝❤ó♥❣ t❛ ❝ơ♥❣ sû ❞ư♥❣ ❣â✐ ♣❤➛♥ ♠➲♠ t♦→♥ ❤å❝ ▼❛♣❧❡ ✶✷ ✈➔ ▼❛t❤❈❛❞ ✶✹✳ ▼❛t❤❈❛❞ ✶✹ ✤➣ ✤÷đ❝ sû ❞ư♥❣ ❝❤♦ ♠ư❝ ✤➼❝❤ t➼♥❤ t♦→♥ ❦❤ỉ♥❣ s❛✐ ❝→❝ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ✣✐➲✉ ♥➔② ❝✉♥❣ ❝➜♣ t❤➯♠ ❦➳t q✉↔ ✹✶ t✐♥ ❝➟② s ợ ộ ữỡ tr r s❛♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❚r♦♥❣ ▼❛♣❧❡ ✶✷ sü ❦✐➸♠ tr❛ ✤➣ ✤÷đ❝ t❤ü❝ ❤✐➺♥✳ ❈❤ó♥❣ t❛ t❤➜② r➡♥❣ ✈✐➺❝ t❤➳ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t❤✉ ✤÷đ❝ ❝õ❛ ❜➲ ♠➦t ♠♦♥❣ ♠✉è♥ ✈➔♦ ♠é✐ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❤➺ t❤è♥❣ ❜❛♥ ✤➛✉✿ ∂F ∂F ∂F ∂F (128x − 57) + (64x + 60) + 128y + 64y + 168x1 + 2 ∂x ∂x ∂y ∂y 2 147x2 − 192F = ∂F ∂F ∂F (16x2 + 15) − 18 + 16y2 − 42x1 − 42x2 = ∂x1 ∂x2 ∂y1 ∂F ∂F ∂F (16x + 15) − 16y − − 6y1 + 42y2 = 2 ∂y ∂x ∂y 1 ∂F + 3y2 = ∂y1 t❤➻ s➩ t❤✉ ✤÷đ❝ ❝→❝ ✤➥♥❣ t❤ù❝ ✤ó♥❣✳ ✹✷ ❑➌❚ ▲❯❾◆ ◗✉❛ ♠ët t❤í✐ ❣✐❛♥ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t➔✐ ❧✐➺✉✱ ❝ị♥❣ ✈ỵ✐ sỹ ữợ t t ổ ❚❤ị② ❉÷ì♥❣ ♥❛② ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ❚♦➔♥ ❦❤â❛ ❧✉➟♥ ✤➣ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✧ ✣↕✐ sè tr÷í♥❣ ✈❡❝t♦r ✈➔ t➼❝❤ ♣❤➙♥ ❝❤ó♥❣✧✳ ❈ư t❤➸ ❧➔ ✤➣ ♥➯✉ ✈➔ ❧➔♠ rã ✈➼ ❞ö ✈➲ ✤↕✐ sè ♠❛ tr➟♥✱ ❝❤✉②➸♥ ✤è✐ s❛♥❣ tå❛ ✤ë t❤ü❝✱ ❧➜② t➼❝❤ ♣❤➙♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ❉♦ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❦❤â❛ ❧✉➟♥ ❝â ❤↕♥ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ữủ õ ỵ t ổ ỗ ❣✐→ ✈➔ t♦➔♥ t❤➸ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ tr♦♥❣ ❦❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣ ✤➸ ❦❤â❛ ❧✉➟♥ ❝â t❤➸ ♣❤→t tr✐➸♥ tèt ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▲♦❜♦❞❛✱ ❆❱ ❖♥ ❛ ❢❛♠✐❧② ♦❢ ❛❢❢✐♥❡✲❤♦♠♦❣❡♥❡♦✉s r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ♦❢ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡① s♣❛❝❡ ✴ ❆❱ ▲♦❜♦❞❛✱ ❆❙ ❑❤♦❞❛r❡✈ ✴ ✴ Pr♦✲ ❝❡❡❞✐♥❣s ♦❢ t❤❡ ✉♥✐✈❡rs✐t✐❡s✳ ❙❡r✳ ▼❛t❤❡♠❛t✐❝s✳ ❬✷❪ ▲♦❜♦❞❛✱ ❆❱ ❖♥ ❛ ❢❛♠✐❧② ♦❢ ▲✐❡ ❛❧❣❡❜r❛s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❍♦♠♦❣❡♥❡♦✉s ❙✉r❢❛❝❡s ✴ ❆❱ ▲♦❜♦❞❛ ✴ ✴ Pr♦❝ ✳ ✲ ✷✵✵✻ ✳ ✲ ❚✳ ✷✺✸✳ ✲ ❙✳ ✶✶✶ ✲ ✶✷✻✳ ❬✸❪ ❉❛♥✐❧♦✈ ✱ ▼❙ ❆❢❢✐♥❡✲ ❤♦♠♦❣❡♥❡♦✉s r❡❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ✐♥❞❡❢✐♥✐t❡ ▲❡✈✐ ❢♦r♠ ✴ ✴ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ♣❛rt✐❝✐♣❛♥ts ♦❢ t❤❡ s❝❤♦♦❧ ✲ s❡♠✐♥❛r ♦♥ ❣❡♦♠❡tr② ❛♥❞ ❛♥❛❧②s✐s ✭ ❆❜r❛✉ ❉✉rs♦ ✮ ✱ ❘♦st♦✈ ✲❉ ✳ ✷✵✵✽ ✱ ♣✳✷✺ ✲✷✻ ✳ ❬✹❪ ▲♦❜♦❞❛✱ ❆❱ ❆❝t✐♦♥ ✐♥ t❤❡ ❝♦♠♣❧❡① ❛❢❢✐♥❡ s✉❜❣r♦✉♣ t❛♥❣❡♥t ♣❧❛♥❡ t♦ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❤♦♠♦❣❡♥❡♦✉s ✴ ❆❱ ▲♦❜♦❞❛ ✴ ✴ ❱♦r♦♥❡③❤ ❲✐♥t❡r ▼❛t❤✳ ❙❝❤♦♦❧ ✳ ❚❡③ ✳ ♦❢ r❡♣♦rts ✳ ✲ ❱♦r♦♥❡③❤ ✷✵✵✾ ✳ ✲ P♣✳ ✶✵✻ ✲ ✶✵✼✳ ❬✺❪ ✣➦♥❣ ◆❣å❝ ❉ö❝ ✲ ◆❣✉②➵♥ ❱✐➳t ✣ù❝✱ ❚♦→♥ ❈❛♦ ❈➜♣ ✭P❤➛♥ ■■ ✣↕✐ ❙è ❚✉②➳♥ ❚➼♥❤✮✱ ◆①❜✳✣➔ ◆➤♥❣✱ ✷✵✵✾✳ ❬✻❪ ◆❣✉②➵♥ ❍♦➔♥❣ ✲ ▲➯ ❱➠♥ ❍↕♣✱ ●✐→♦ ❚r➻♥❤ ●✐↔✐ ❚➼❝❤ ❍➔♠✱ ◆①❜✳❍✉➳✱ ✶✾✾✼✳ ✹✹ ... ❦❤→❝ ♥❤❛✉ ❦❤✐ t❤❛② ✤ê✐ ❝→❝ t➼❝❤ ▲✐❡ ❦❤→❝ ♥❤❛✉✳ ✻ ▼é✐ ✤↕✐ sè ▲✐❡ ❧➔ ♠é✐ ❑●❱❚ ♥➯♥ sè ❝❤✐➲✉ ❝õ❛ ✤↕✐ số số ã Pữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣ ❈â ♥❤✐➲✉ ❜➔✐ t♦→♥ tỹ t t ỵ tt tỵ✐ ✈✐➺❝ t➻♠ ♠ët ❤➔♠... ✤↕✐ sè ✤â ❧➔ ♠ët ✈✐➺❝ ❦❤â ❦❤➠♥✳ ▼ö❝ t✐➯✉ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ①➙② ❞ü♥❣ ❜➲ ♠➦t ỗ t t ỳ số trữợ số tữỡ ự ợ trữớ ủ t ổ ①→❝ ✤à♥❤✳ ✷✳✷ ❈❤✉②➸♥ ✤ê✐ s❛♥❣ tå❛ ✤ë t❤ü❝ ✣✐➲✉ ❦✐➺♥ t✐➳♣ ①ó❝ tr÷í♥❣... t❤✉ ✤÷đ❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ s tỹ ỗ t ổ ự ✸ ❝❤✐➲✉ ✭①❡♠ ❝→❝ t➔✐ ❧✐➺✉ ✶✲✹✮✳ ❇➜t ❦➻ ✤↕✐ số ỗ tữỡ ự ợ ởt t ỗ ♥❤➜t ❛❢❢✐♥❡ ♥➔♦ ✤â✳ ▲➔♠ t❤➳ ♥➔♦ ✤➸ ❝â ✤÷đ❝ ✤↕✐ sè ✈➔ ①➙② ❞ü♥❣ ❜➲ ♠➦t t÷ì♥❣