Tìm hiểu về định lý bốn đỉnh

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❷◆● ❇➐◆❍ ❑❍❖❆ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✣❖⑨◆ ◗❯Ý◆❍ ❍❸◆❍ ❚➐▼ ❍■➎❯ ❱➋ ✣➚◆❍ ▲Þ ❇➮◆ ✣➓◆❍ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ◗✉↔♥❣ ❇➻♥❤✱ t❤→♥❣ ✺✱ ♥➠♠ ✷✵✶✾ ✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✱ ❝→❝ sè ❧✐➺✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❣❤✐ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❧➔ tr✉♥❣ t❤ü❝✱ ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ ❦❤â❛ ❧✉➟♥ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❦❤â❛ ❧✉➟♥✱ tæ✐ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ◗✉↔♥❣ ❇➻♥❤✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✣♦➔♥ ◗✉ý♥❤ ❍↕♥❤ ✐✐ ▲❮■ ❈❷▼ ❒◆ ❚r♦♥❣ s✉èt t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✱ ♥❣♦➔✐ sü ♥é ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥✱ tæ✐ ❝á♥ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉↔♥❣ ❇➻♥❤✳ ✣➦❝ ❜✐➺t tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❣✐→♦ ❚❤s✳ ◆❣✉②➵♥ ▲➯ ❚r➙♠✳ ❚❤➛② ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ qỵ t t ữợ tổ tr sốt q tr tỹ õ tốt ỗ tớ ú tỉ✐ ❧➽♥❤ ❤ë✐ ✤÷đ❝ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❝❤✉②➯♥ ♠ỉ♥ ✈➔ r➧♥ ❧✉②➺♥ ❝❤♦ tæ✐ t→❝ ♣❤♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ◗✉❛ ✤➙② tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥✱ ✈➔ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ơ♥❣ ♥❤÷ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ▼➦❝ ❞ị ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ♥❤÷♥❣ ♥ë✐ ❞✉♥❣ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ tổ rt ữủ sỹ õ ỵ ❝õ❛ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ◗✉↔♥❣ ❇➻♥❤✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✣♦➔♥ ◗✉ý♥❤ ử P ỵ ❜è♥ ✤➾♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ✶✳✶ ✣÷í♥❣ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✤÷í♥❣ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ✣÷í♥❣ t❤❛♠ sè ❝❤➼♥❤ q✉②✳ ✣ë ❞➔✐ ❝✉♥❣ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✤÷í♥❣ t❤❛♠ sè tr♦♥❣ R3 ✶✳✷✳✶ ✣ë ❝♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❚r÷í♥❣ ♠ư❝ t✐➯✉ ❋r➨♥❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ✣ë ①♦➢♥✳ ❈æ♥❣ t❤ù❝ ❋r➨♥❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✹ ❈æ♥❣ t❤ù❝ t➼♥❤ ✤ë ❝♦♥❣ ✈➔ ✤ë ①♦➢♥ ỵ ❝ì ❜↔♥ ❝❤♦ ✤÷í♥❣ t❤❛♠ sè tr♦♥❣ R3 ✳ ✳ ✶✳✸ ✣÷í♥❣ t❤❛♠ sè tr♦♥❣ R2 ✭✣÷í♥❣ t❤❛♠ sè ♣❤➥♥❣✮ ỵ ố ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✺ ✺ ✺ ✽ ✶✵ ✶✵ ✶✶ ✶✷ ✶✺ ✶✼ ✶✽ ✷✶ ✷ ỵ ố tr ổ ợ t ❧♦❣✲t✉②➳♥ t➼♥❤ ✷✼ ✷✳✶ ❑❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ❑❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ▼➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣à♥❤ ỵ ố tr ổ ợ t t t ỵ ố tr ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❈→❝ ✤÷í♥❣ ❝♦♥❣ ❝â f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ey ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✷✼ ✷✼ ✷✽ ✷✽ ✷✾ ✸✶ ✹✵ ✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✷ P❤➛♥ ♠ð ✤➛✉ ❚r♦♥❣ ❍➻♥❤ ❤å❝ ✈✐ ♣❤➙♥ ❝ê ✤✐➸♥✱ ♥❤➔ ❚♦→♥ ❤å❝ ▼✳ P✳ ❞♦ ❈❛r♠♦ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❝❤♦ ❝❤ó♥❣ t❛ ❜✐➳t ✤÷đ❝ ♥❤ú♥❣ ✤➦❝ ✤✐➸♠✱ t➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ✈➔ ♠➦t ♣❤➥♥❣✳ ❚r♦♥❣ ✤â ❝â ♠ët t➼♥❤ ❝❤➜t t♦➔♥ ❝ư❝ r➜t t❤ó ✈à ✤â ❝❤➼♥❤ ❧➔ ✣à♥❤ ỵ ố ỵ ố tr ✈✐ ♣❤➙♥ ❝ê ✤✐➸♥ ❝❤♦ t❛ ❜✐➳t r➡♥❣✿ ❚r♦♥❣ ❦❤æ♥❣ ợ t t ởt ữớ õ ỡ ỗ õ t t ố ❑❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ Rn ❝ị♥❣ ợ ởt trỡ ữỡ ỵ ef ✱ ❞ò♥❣ ✤➸ ❧➔♠ trå♥❣ sè ✤➸ ✤→♥❤ ❣✐→ ✤ë ❞➔✐✱ ❞✐➺♥ t➼❝❤✱ t❤➸ t➼❝❤✱ ✳✳✳ ✣➸ ♣❤➙♥ ❜✐➺t ✈ỵ✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✤ë ❞➔✐✱ ❞✐➺♥ t➼❝❤✱ t❤➸ t➼❝❤✱ ✳✳✳ t❤ỉ♥❣ t❤÷í♥❣✱ ❝→❝ ✤ë ❞➔✐✱ ❞✐➺♥ t➼❝❤✱ t❤➸ t➼❝❤✱ ✳✳✳ ợ t ữủt ữủ f ❞➔✐✱ f ✲❞✐➺♥ t➼❝❤✱ f ✲t❤➸ t➼❝❤✱ ✳✳✳ ❑❤æ♥❣ ❣✐❛♥ Rn ợ t ef ữủ ỵ ④ Rn , e−f ⑥ ✳ ❍➔♠ f ❝â ❞↕♥❣ f (x1, x2, , xn) = aixi t❤➻ ♠➟t ✤ë ữủ t t t t ợ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ❝â f = ax + by✳ ❈â ❤❛✐ ✈➜♥ ✤➲ ✤÷đ❝ ✤➦t r❛ ✤â ❧➔✿ ✣à♥❤ ỵ ố tr ❝â ❝á♥ ✤ó♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t ỳ ổ ỵ ố ổ ỏ ú t ợ ữớ tr ổ ợ t t t tọ ỵ ố ợ ỵ tr ữợ sỹ ủ ỵ ữợ t ▲➯ ❚r➙♠✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ õ ỵ ố ◆❣♦➔✐ ❝→❝ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ❝❤✉♥❣ ✈➔ ❦✐➳♥ ♥❣❤à✱ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✿ ✣à♥❤ ỵ ố tr ổ r s ữớ t số ã t t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✤÷í♥❣ t❤❛♠ sè tr♦♥❣ R3 ✭✣ë ❝♦♥❣✱ tr÷í♥❣ ♠ư❝ t✐➯✉ ❋r➨♥❡t✱ ✤ë ①♦➢♥✱ ❝ỉ♥❣ t❤ù❝ ❋r➨♥❡t✱ ✤à♥❤ ỵ ỡ ữớ t số tr R3 ã ữớ t số tr R2 ã ỵ ố tr ổ ã ữỡ ỵ ❜è♥ ✤➾♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤✳ ❚r➻♥❤ ❜➔② ❝→❝ ♥ë✐ ❞✉♥❣ s❛✉✿ ❑❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t t ợ t ã t ỵ ố õ ỏ ú tr ổ ợ t t t ổ ợ ữớ ❝♦♥❣ ❝â f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ t ey ã ữỡ ỵ ❜è♥ ✤➾♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ❑✐➳♥ t❤ù❝ ❝õ❛ ♠ư❝ ♥➔② tỉ✐ tê♥❣ ❤đ♣ tø ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❤å❝ ❝õ❛ ❤å❝ ♣❤➛♥ ❍➻♥❤ ❤å❝ ✈✐ ♣❤➙♥ ✈➔ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉ ❬✺❪ ✶✳✶ ✣÷í♥❣ t❤❛♠ sè ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✤÷í♥❣ t❤❛♠ sè ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ →♥❤ ①↕ c : I −→ Rn ✈ỵ✐ I ⊂ R ❧➔ ♠ët ❦❤♦↔♥❣ ✭♠ð✱ ✤â♥❣✱ ♥û❛ ♠ð ♥û❛ ✤â♥❣✱ ♥û❛ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ ❤♦➦❝ ❝↔ t♦➔♥ ❜ë ✤÷í♥❣ t❤➥♥❣ t❤ü❝✳✳✳✮✳ ●å✐ C = c(I ) ⊂ Rn✱ ↔♥❤ ❝õ❛ t♦➔♥ ❜ë t➟♣ I ✳ ❑❤✐ ✤â (C, c) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ữớ t số rtr r ợ t số õ c ✈➔ t❤❛♠ sè t✳ C ✤÷đ❝ ❣å✐ ❧➔ ✈➳t ❝õ❛ ✤÷í♥❣ t❤❛♠ sè✳ ◆➳✉ c ❧➔ ❤➔♠ ❧✐➯♥ tư❝✱ ❦❤↔ ✈✐ ❧ỵ♣ C k ✱ ❦❤↔ ✈✐ ❧ỵ♣ C ∞✳✳✳ t❤➻ t÷ì♥❣ ù♥❣ t❛ ♥â✐ C ❧➔ ✤÷í♥❣ t❤❛♠ sè ❧✐➯♥ tư❝✱ ❦❤↔ ✈✐ ❧ỵ♣ C k ✱ ❦❤↔ ✈✐ ❧ỵ♣ C ∞ ✳✳✳ ●✐↔ sû c(t) = (x1(t), x2(t), xn(t)))✱ t❤➻ c ❦❤↔ ✈✐ ❧ỵ♣ C k (k = 0, 1, 2, ) ❝â ♥❣❤➽❛ ❧➔ ❝→❝ ❤➔♠ t❤➔♥❤ ♣❤➛♥ xi : I −→ R ✺ ❦❤↔ ✈✐ ❧ỵ♣ C k (k = 0, 1, 2, ) ◆➳✉ c ❧➔ ❦❤↔ ✈✐ t❤➻ ✈❡❝t♦r c (t) := x1(t), x2(t), , xn(t)) ∈ Rn✱ ❣å✐ ❧➔ ✈❡❝t♦r t✐➳♣ ①ó❝ ❝õ❛ C t c(t) c t t ú ỵ ✶✳✶✳✶✳ ✶✳ ❚r♦♥❣ s✉èt ❦❤â❛ ❧✉➟♥ ♥➔②✱ ♥➳✉ ❦❤æ♥❣ ♥â✐ ❣➻ t❤➯♠✱ t❤✉➟t ♥❣ú ❦❤↔ ✈✐ ✤÷đ❝ ❤✐➸✉ ❧➔ ❦❤↔ ✈✐ t↕✐ ♠å✐ ✤✐➸♠ ✈➔ ❦❤↔ ✈✐ ✤➳♥ ❧ỵ♣ ❝➛♥ t❤✐➳t✳ ❚ø ✤➙② trð ✤✐ ❝❤ó♥❣ t❛ ❝❤➾ ①➨t ❝→❝ ✤÷í♥❣ t❤❛♠ sè ❦❤↔ ✈✐✳ ❱➻ t❤➳✱ ❦❤✐ ❦❤ỉ♥❣ ❝➛♥ ♥❤➜♥ ♠↕♥❤ ❝❤ó♥❣ t❛ s➩ ❜ä ✤✐ tø ❦❤↔ ✈✐✳ ỡ t ũ ỵ ✤õ (C, c) ✤➸ ❝❤➾ ✤÷í♥❣ t❤❛♠ sè t❛ ❝â t❤➸ ♥â✐ C ❧➔ ✤÷í♥❣ t❤❛♠ sè ♥➳✉ t❤❛♠ sè ❤â❛ ✤➣ ❜✐➳t✳ ❚❤➟t r❛ t❤❛♠ sè ❤â❛ ❝õ❛ ✤÷í♥❣ t❤❛♠ sè ❝❤♦ ♣❤➨♣ t❛ ①→❝ ✤à♥❤ ✈➳t ❝õ❛ ♥â ♥➯♥ ❦❤✐ ♥â✐ ✈➲ ✤÷í♥❣ t❤❛♠ sè ❝❤➾ ❝➛♥ ❝❤♦ t❤❛♠ sè ❤â❛ ❝õ❛ ♥â ❧➔ ✤õ✳ ❚❛ t❤è♥❣ ♥❤➜t ữớ t số ợ t số õ ✤÷í♥❣ ❝♦♥❣ s➩ ✤÷đ❝ ❤✐➸✉ ❧➔ ✈➳t ❝õ❛ ♠ët ✤÷í♥❣ t❤❛♠ sè ♥➔♦ ✤â✳ ✹✳ ▼ët t➟♣ ❝♦♥ C ⊂ Rn ❝â t❤➸ ❝â ♥❤✐➲✉ t❤❛♠ sè ❤â❛ ❦❤→❝ ♥❤❛✉✳ ữớ t số ợ t số õ c(t) = p + tv, p, v ∈ Rn , v = 0, t ∈ R ❧➔ ✤÷í♥❣ t❤➥♥❣ ✤✐ q p ợ tr ữỡ v ✶✳✶✳✷✳ ✣÷í♥❣ trá♥ t➙♠ O✱ ❜→♥ ❦➼♥❤ r ❝â ♠ët t❤❛♠ sè ❤â❛ ❞↕♥❣ c(t) = (r cos(t), r sin(t)) ❍➻♥❤ ✶✳✶✿ c(t) = (t, f (t)) ❍➻♥❤ ✶✳✷✿ c(t) = (x(t), y(t)) ❱➼ ❞ư ✶✳✶✳✸✳ ✣÷í♥❣ ♣❛r❛❜♦❧ ❝â ♠ët t❤❛♠ sè ❤â❛ ❞↕♥❣ c(t) = (t, t2 ) ữớ t số C ợ t sè ❤â❛ ✻ c(t) = (a cos t, a sin t, bt) ; t ∈ R, a > 0, b = ✣÷í♥❣ t❤❛♠ sè C ❣å✐ ❧➔ ✤÷í♥❣ ①♦➢♥ è❝✳ ✣÷í♥❣ ♥➡♠ tr➯♥ ♠➦t trư x2 + y2 = a2 ✈ỵ✐ ✤ë ❞è❝ 2πb✳ ❚❤❛♠ sè t ❝❤➼♥❤ ❧➔ õ ỳ trử x ợ ữớ t ố O ợ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ c(t) ❧➯♥ ♠➦t ♣❤➥♥❣ Oxy✳ ❱➼ ❞ö ✶✳✶✳✺✳ ⑩♥❤ ①↕ c : R → R2✱ ①→❝ ✤à♥❤ ❜ð✐ c(t) = (t3 , t2 ) ; t ∈ R✱ ❧➔ t❤❛♠ sè ❤â❛ ❝õ❛ ♠ët ✤÷í♥❣ t❤❛♠ sè ợ C ú ỵ r c (0) = (0, 0)✱ tù❝ ❧➔ t↕✐ t = ✈❡❝t♦r t✐➳♣ ①ó❝ ❜➡♥❣ 0✳ ❱➼ ❞ư ✶✳✶✳✻✳ ⑩♥❤ ①↕ c : R → R2✱ ①→❝ ✤à♥❤ ❜ð✐ c(t) = (t3 − 4t, t2 − 4) ; t ∈ R✱ ❧➔ t❤❛♠ sè ❤â❛ ❝õ❛ ♠ët ✤÷í♥❣ t❤❛♠ sè ❦❤↔ ✈✐ ợ C ú ỵ r c(2) = c(2) = (0, 0)✱ tù❝ ❧➔ →♥❤ ①↕ c ❦❤ỉ♥❣ ✤ì♥ →♥❤✳ ❱➼ ❞ö ✶✳✶✳✼✳ ⑩♥❤ ①↕ c : R → R2✱ ①→❝ ✤à♥❤ ❜ð✐ c(t) = (t, |t|) ; t ∈ R✱ ❧➔ t❤❛♠ sè ❤â❛ ❝õ❛ ♠ët ✤÷í♥❣ t❤❛♠ sè ❧✐➯♥ tư❝ ❦❤ỉ♥❣ ❦❤↔ ✈✐ ✈➻ ❤➔♠ y(t) = |t| ❦❤ỉ♥❣ ❦❤↔ ✈✐ t↕✐ t = 0✳ ❱➼ ❞ư ✶✳✶✳✽✳ ❍❛✐ →♥❤ ①↕ c, r : R → R2✱ ①→❝ ✤à♥❤ ❜ð✐ c(t) = (cos t, sin t) , r(t) = (cos 2t, sin 2t) ; ❧➔ ❤❛✐ t❤❛♠ sè ❤â❛ ❦❤→❝ ♥❤❛✉ ❝õ❛ ✤÷í♥❣ trá♥ x2 + y2 = ú ữớ t số ợ ✈❡❝t♦r t✐➳♣ ①ó❝ t↕✐ tø♥❣ ✤✐➸♠ ❧➔ ❦❤→❝ ♥❤❛✉ ✈➻ ❝â ✤ë ❞➔✐ ❦❤→❝ ♥❤❛✉✳ ❱➼ ❞ö ✶✳✶✳✾✳ ❍❛✐ →♥❤ ①↕ c, r : R → R2✱ ①→❝ ✤à♥❤ ❜ð✐ c(t) = (t, t) , r(t) = t3 , t3 ; ❧➔ ❤❛✐ t❤❛♠ sè ❤â❛ ❝õ❛ ❝ị♥❣ ♠ët ✤÷í♥❣ t❤➥♥❣ x = y✳ ❈❤ó♥❣ ①→❝ ✤à♥❤ ❤❛✐ ✤÷í♥❣ t❤❛♠ sè ✈ỵ✐ ❝→❝ ✈❡❝t♦r t✐➳♣ ①ó❝ t↕✐ tø♥❣ ✤✐➸♠ ❧➔ ❦❤→❝ ♥❤❛✉✳ ❍❛✐ ✤÷í♥❣ ❝♦♥❣ ♥➔② ♠ỉ t↔ ❤❛✐ ❝❤✉②➸♥ ✤ë♥❣ ❝ị♥❣ q✉ÿ ✤↕♦ ♥❤÷♥❣ ❝→❝❤ ❝❤✉②➸♥ ✤ë♥❣ ❤♦➔♥ t♦➔♥ ❦❤→❝ ♥❤❛✉✳ ✣÷í♥❣ ❝♦♥❣ t❤ù ♥❤➜t ♠ỉ t↔ ❝❤✉②➸♥ ✤ë♥❣ ✤➲✉ tr➯♥ ✤÷í♥❣ t❤➥♥❣✳ ✣÷í♥❣ ❝♦♥❣ t❤❛♠ sè t❤ù ❤❛✐ ♠ỉ t↔ ❝❤✉②➸♥ ✤ë♥❣ ❝❤➟♠ ❞➛♥ ✭✈ỵ✐ t < 0✮✱ ✈➟♥ tè❝ tù❝ t❤í✐ ❜➡♥❣ ❦❤ỉ♥❣ t↕✐ t = 0✱ ✈➔ s❛✉ ✤â ✭✈ỵ✐ t > 0✮ ❝❤✉②➸♥ ✤ë♥❣ ♥❤❛♥❤ ❞➛♥✳ ✼ ✹✳ ❍➔♠ f ❝â ❞↕♥❣ f (x1, x2, , xn) = ϕ ( ❧♦❣✲❜➟❝ ❤❛✐❀ x2i ) ợ t t ữủ t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ✶✳ ▼➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤✿ f = ax + by✳ ✷✳ ▼➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ s✉② rë♥❣✿ f = ϕ(ax + by)✳ ✸✳ ▼➟t ✤ë ❝➛✉✿ f = ϕ(r)✳ ✹✳ ▼➟t ✤ë ❧♦❣✲❜➟❝ ❤❛✐✿ f = ϕ(ax2 + by2)✳ Ð ✤➙② ❝❤ó♥❣ tỉ✐ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ✈ỵ✐ f = ax + by✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✸✳ ❬✻❪ ❚r➯♥ ♠➦t ♣❤➥♥❣ ✱ ❝❤♦ ✤÷í♥❣ t❤❛♠ sè α : I → R2✱ ✤ë ❝♦♥❣ ✤↕✐ sè t❤❡♦ ♠➟t ✤ë ❤❛② f ✲✤ë ❝♦♥❣✱ ❦➼ ❤✐➺✉ kf ✱ ❝õ❛ α ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ df = k + < ∇f, n > ✭✷✳✶✮ kf = k + dn tr♦♥❣ ✤â k ❧➔ ✤ë ❝♦♥❣ ✤↕✐ sè ✈➔ n ❧➔ tr÷í♥❣ ✈❡❝t♦r ♣❤→♣ ✤ì♥ ✈à ❞å❝ ✤÷í♥❣ ❝♦♥❣ α✳ ❈ư t❤➸✱ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ✈ỵ✐ f = ax + by t❤➻ R2 ; e−f ∂f ∂f + =a+b ∂x ∂y =⇒ kf = k + (a + b).n ∇f = ❇ê ✤➲ ✷✳✶✳✶✳ ❬✻❪ ❚r➯♥ ♠➦t ♣❤➥♥❣ ✱ ❝❤♦ ✤÷í♥❣ t❤❛♠ sè α(s) : (x(s), y(s)) ✈ỵ✐ s ❧➔ t❤❛♠ sè ✤ë ❞➔✐ ❝✉♥❣✳ ❑❤✐ ✤â✱ f ✲✤ë ❝♦♥❣ ❝õ❛ α ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤➥♥❣ t❤ù❝ ∂f ∂f kf = x y − x y − y +x ✭✷✳✷✮ ∂x ∂y R2 ; e−f ❈❤ù♥❣ ♠✐♥❤✳ ❈ỉ♥❣ t❤ù❝ tr➯♥ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø ✭✷✳✶✮ ✈➔ ✭✶✳✺✮ ✈ỵ✐ s ❧➔ t❤❛♠ sè ✤ë ❞➔✐ ỵ ố tr ổ ợ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ❚r♦♥❣ ♣❤➛♥ ♥➔② tæ✐ t➟♣ tr✉♥❣ qt ỵ ố ✤➾♥❤ ❝â ❝á♥ ✤ó♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë t t ổ ỵ ố ổ ỏ ú ợ ữớ tr ổ ợ t t t tọ ỵ ố ỵ ố tr ổ ợ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë t✉②➳♥ t➼♥❤✱ ♠➔ t❤❡♦ tr➯♥ t❛ ❣✐↔ sû ♠➟t ✤ë ú t õ ỵ ố ởt ợ ữớ tọ t ởt ✤è✐ ①ù♥❣✳ ❉♦ e−f (x) kf = k + < ∇f, n > ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ t➟♣ ❝♦♠♣❛❝t [a, b] ♥➯♥ kf ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ♥❤ä ♥❤➜t tr➯♥ [a, b]✳ ❚ø ✤â ❝❤ó♥❣ t❛ ❝â ✣à♥❤ ❧➼ ✷✳✷✳✶✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ (R2; e−f (x))✱ ♠å✐ ✤÷í♥❣ ❝♦♥❣ ✤ì♥ ✤â♥❣ ❝â ➼t ♥❤➜t ❤❛✐ ✤➾♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû α : [a,b] −→ R2 θ −→ α(θ) ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ✤ì♥ ✤â♥❣✳ ❑❤✐ ✤â✱ ✤ë ❝♦♥❣ kf ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ t❤❡♦ θ tr➯♥ t➟♣ ❝♦♠♣❛❝t [a, b] ♥➯♥ kf ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ♥❤ä ♥❤➜t t↕✐ θ1, θ2 ∈ [a, b]✳ ❉♦ ✤÷í♥❣ ❝♦♥❣ α ❧➔ ✤â♥❣ ♥➯♥ ❤❛✐ ✤✐➸♠ α(θ1) ✈➔ α(θ2) ❧➔ ❤❛✐ ✤➾♥❤ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ α✳ ✣à♥❤ ❧➼ ✷✳✷✳✷✳ ❈❤♦ (C ) ❧➔ ✤÷í♥❣ trá♥ t➙♠ (a, 0)✱ ❜→♥ ❦➼♥❤ R tr♦♥❣ ♠➦t ♣❤➥♥❣ (R2 ; e−f (x) ) ✈➔ α : [0, 2π ] → R2 ✱ α(t) = (R cos t + a; R sin t) ❧➔ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ ❝❤ó♥❣✳❈❤ó♥❣ t❛ ❝â✿ dkf y =− f + f (x − a) dt R ✷✾ ✭✷✳✸✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â α(t) = (R cos t + a; R sin t) , α (t) = (−R sin t, R cos t) , ∇f = (f , 0), α (t) = (−R cos t, −R sin t) ✣ë ❝♦♥❣ t❤❡♦ ♠➟t ✤ë e−f ❝õ❛ ✤÷í♥❣ trá♥ ❧➔✿ kf = k + < ∇, n >= R + R fy ❉♦ ✤â dkf = f y +f xy dt R = f (−y ) + f (x − a)(−y ) R =− y f + f (x − a) R ✭✷✳✹✮ ✭✷✳✺✮ ✭✷✳✻✮ ❈❤ó♥❣ t❛ ❝â ❝→❝ ❤➺ q✉↔ s❛✉✿ ❍➺ q✉↔ ✷✳✷✳✶✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ (R2; e−f (x)) ✶✳ ❱ỵ✐ f (x) = Ax + B, A, B ∈ R, A = t❤➻ ♠å✐ ✤÷í♥❣ trá♥ ✤➲✉ ❝â ✤ó♥❣ ❤❛✐ ữớ trỏ õ ợ t ✤ë ❤➡♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❧➔ ❤➡♥❣✳ ❍➺ q✉↔ s❛✉ ❝❤♦ t❛ ✤✐➲✉ ❦✐➺♥ ✈➲ ♠➟t ✤ë ✤➸ ✤÷í♥❣ trá♥ t➙♠ (a, 0) ❜→♥ ❦➼♥❤ R ❝â ✤ë ❝♦♥❣ ✈ỵ✐ ♠➟t ✤ë ❤➡♥❣✳ ❍➺ q✉↔ ✷✳✷✳✷✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ (R2; e−f (x))✱ ✤÷í♥❣ trá♥ t➙♠ (a, 0)✱ ❜→♥ ❦➼♥❤ R ❝â ✤ë ❝♦♥❣ ✈ỵ✐ ♠➟t ✤ë ❤➡♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) = c ln |x − a| + b ✈ỵ✐ b, c ❧➔ ❤➡♥❣ sè ❜➜t ❦➻✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣÷í♥❣ trá♥ t➙♠ (a, 0) ❜→♥ ❦➼♥❤ R ❝â ✤ë ❝♦♥❣ ✈ỵ✐ ♠➟t ✤ë ❤➡♥❣ ❦❤✐ ✈➔ kf = tữỡ ữỡ ợ f + f ( x − a) = ⇐⇒ (x − a)f =0 ∀x ∈ (a − R; a + R) ∀x ∈ (a − R; a + R) ⇐⇒(x − a)f = c c ⇐⇒f = x−a ✸✵ ✣à♥❤ ❧➼ ✷✳✷✳✸✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ R2 ✈ỵ✐ ♠➟t ✤ë e−ax−by ✱ ✤÷í♥❣ trá♥ (C ) ❝â t➙♠ I (a, b), a, b ∈ R✱ a = b = 0✳ ❜→♥ ❦➼♥❤ R > ❝â ❤❛✐ ✤➾♥❤ ♥➳✉ a2 + b2 = ✈➔ ❝â ✈æ sè ✤➾♥❤ ♥➳✉ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t❤❛♠ sè ❤â❛ tü ♥❤✐➯♥ ❝õ❛ ✤÷í♥❣ trá♥ (C ) ❝â ❞↕♥❣ α : [0,2π ] −→ R2 t t −→ R cos( ) + c, R sin( ) + d R R t ⑩♣ ❞ö♥❣ ❝ỉ♥❣ t❤ù❝ ✭✷✳✷✮✱ t❛ t➼♥❤ ✤÷đ❝ kf = t t − a cos( ) + b sin( ) R R R ✣➥♥❣ t❤ù❝ tr➯♥ s✉② r❛ ✤÷đ❝ kf = t t a sin( ) + b cos( ) R R R ❉♦ ✤â✱ ♥➳✉ a = b = t❤➻ kf ❧➔ ❤➡♥❣ sè✳ ❑❤✐ ✤â✱ ✤÷í♥❣ trá♥ (C ) ❝â ✈æ sè ✤➾♥❤✳ ◆➳✉ a2 + b2 = t❤➻ ♣❤÷ì♥❣ tr➻♥❤ kf = ❝â ✤ó♥❣ ❤❛✐ ♥❣❤✐➺♠✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ✤÷í♥❣ trá♥ (C ) ❝â ✤ó♥❣ ❤❛✐ ✤➾♥❤✳ ✷✳✷✳✷ ❈→❝ ✤÷í♥❣ ❝♦♥❣ ❝â f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ey ❈❤ó♥❣ tỉ✐ t✐➳♥ ❤➔♥❤ ♣❤➙♥ ❧♦↕✐ ❝→❝ ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ey ỵ r f ữớ t sè α tr➯♥ ♠➦t ♣❤➥♥❣ ♥➔② ❜➜t ❜✐➳♥ q✉❛ ❝→❝ ♣❤➨♣ tà♥❤ t✐➳♥✳ ❉♦ ✤â✱ ✤➸ ✤ì♥ ❣✐↔♥ tr♦♥❣ tr➻♥❤ ❜➔②✱ ❝❤ó♥❣ tỉ✐ ❜ä q✉❛ ❝→❝ ❤➡♥❣ sè ❦❤✐ t➼♥❤ ❝→❝ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ t❤❛♠ sè ❤â❛ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ α✳ ●✐↔ sû α(s) = (x(s), y(s)) ❧➔ ♠ët ✤÷í♥❣ t❤❛♠ sè tü ♥❤✐➯♥✱ ❝â f ✲✤ë ❝♦♥❣ ❜➡♥❣ ❤➡♥❣ sè c✳ ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ ✤÷đ❝ xy −x y −x =c ✭✷✳✼✮ ❈❤ó♥❣ t❛ ✤➦t x = − cos(2ξ ) ✭✷✳✽✮ y = sin(1ξ ) P❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮ trð t❤➔♥❤ −2ξ + cos(2ξ ) = c ✸✶ ✭✷✳✾✮ cos(2) = c ợ s t ữỡ tr ữủ t ữợ = c + (c + 1) tan2 ξ + tan2 ξ ✭✷✳✶✵✮ ❈❤ó♥❣ t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ ✷✳✷✳✷✳✶✳ ❚r÷í♥❣ ❤đ♣ c < −1 P❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮ ❝â t❤➸ t ữợ 2d tan = (c + 1)ds✱ tan2 ξ + a2 ð ✤â a = c−1 c+1 ❉♦ ✤â✱ − arctan a tan ξ a ❍❛② = (c + 1)s √ tan ξ = a tan c2 + ✭✷✳✶✶✮ s ❇ð✐ ✈➻ c + < 0✳ ❚ø ✤➥♥❣ t❤ù❝ ✭✷✳✶✶✮✱ ❝❤ó♥❣ t❛ ❝â √  c2 +  tan2  s −1 a      √ x (s ) =   c2 +   s + a2 tan2  √ c2 +   a tan s      √ y (s ) =   c2 +    + a2 tan s ❚➼♥❤ t♦→♥ trü❝ t✐➳♣ ❝❤ó♥❣ t❛ ✤↕t ✤÷đ❝ √  c2 −  tan2  − a s    −2   √ x (s ) = √  2−1  c c2 −   + a2 tan2 s  √ c2 −   tan s d    −4   √ y (s) =   ( c + 1) c2 −    + a2 tan2 s ✸✷ √ d tan + tan2 c2 − s √ c2 − √ c −1 tan s √ c2 − + tan s s ✭✷✳✶✷✮ ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✶✷✮ ❝❤ó♥❣ t❛ ✤÷đ❝   (a2 + 1) 4a   x ( s ) = s− √ arctan   (a − 1)  c2 − 1(a2 − 1)    √   tan     ln  y (s ) =   (c + 1)(a − 1)  c −    tan2  c+1 ❚❤❛② a = c−1 c+1 c−1 tan c+1 c2 − √  s c2 − √ c2 − +1 s +1     s ✭✷✳✶✸✮ ✈➔♦ ✭✷✳✶✸✮✱ ❝❤ó♥❣ t❛ ❝â  √  c − c2 +   x ( s ) = − arctan tan s − cs   c+1      √  2−1 c s +1   tan      √ y (s) = − ln       c−1 c −1   tan s +1  c+1 ✷✳✷✳✷✳✷✳ ❚r÷í♥❣ ❤đ♣ c = tỗ t s0 s cos [2(s0)] = −1 t❤➻ ξ(s) = ξ(s0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✾✮✳ ❑❤✐ ✤â α ❧➔ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ✈ỵ✐ trư❝ Ox✳ ◆➳✉ cos [2ξ(s0)] = −1 ✈ỵ✐ ♠å✐ s t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮ trð t❤➔♥❤ ξ = 1 + tan2 ξ ❉♦ ✤â✱ d(tan ξ ) = ds ❚ø ✤â ❝❤ó♥❣ t❛ s✉② r❛ ✤÷đ❝ tan ξ = s ❑❤✐ ✤â t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮✱ ❝❤ó♥❣ t❛ ❝â   x (s) = − − s + s2  y (s) = 2s + s2 ✸✸ ✭✷✳✶✹✮ ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✶✹✮✱ ❝❤ó♥❣ t❛ ✤↕t ✤÷đ❝ ✭✷✳✶✺✮ x(s) = −2 arctan s + s y (s) = ln(1 + s2 ) ✷✳✷✳✷✳✸✳ ❚r÷í♥❣ ❤đ♣ −1 < c < ◆➳✉ tỗ t s0 s cos [2(s0)] = c t ξ(s) = ξ(s0) ❧➔ ♥❣❤✐➺♠ ❞✉② √ ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✾✮✳ ❑❤✐ ✤â✱ α ❧➔ ✤÷í♥❣ t❤➥♥❣ ❝â ❤➺ sè ❣â❝ ❜➡♥❣ − c− c ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ c = 0✱ ❤➺ sè ❣â❝ ❝õ❛ α ữớ t s s ợ trử Oy ữủ cos [2(s0)] = c ợ s t ữỡ tr tữỡ ữỡ ợ ữỡ tr➻♥❤ −d 2d(tan ξ ) = (c + 1)ds, tan2 ξ − b2 ð ✤â b = 1−c =0 c+1 ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✻✮ t❛ ✤÷đ❝ b ln tan ξ + b = (c + 1)s tan ξ − b ❍❛② √ tan ξ + b = e 1−c s tan ξ − b ❈❤ó♥❣ t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ ♥❤ä s❛✉✿ √ 1−c s √ tan ξ + b e ❚r÷í♥❣ ❤đ♣ ❛✿ tan ξ − b = e 1−c s ❤❛② tan ξ = b e√1−c s +− 11 ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❝❤ó♥❣ t❛ ❝â 2  √   e 1−c s +   1− b √    e 1−c s −   x ( s ) = −   √  1−c2 s +   e   1+ b √  e 1−c s − √   e 1−c s +   b √   e 1−c s −    y ( s ) =  √   1−c2 s +  e   1+ b √   e 1−c s − ✸✹ 2 ✭✷✳✶✻✮ ❍❛②  √ √ )e2 1−c2 s − 2(b2 + 1)e2 1−c2 s + − b2  (1 − b   √ √ x (s) = − (1 + b2 )e2 1−c2 s +√2(b2 − 1)e2 1−c2 s + b2 +  2b(e2 1−c s − 1)   √ √ y (s) = (1 + b2 )e2 1−c2 s + 2(b2 − 1)e2 1−c2 s + b2 + ❚❤❛② b = 1−c ✈➔♦ ✭✷✳✶✼✮✱ ❝❤ó♥❣ t❛ ✤÷đ❝ 1+c  2c 2√1−c2 s √1−c2 s 2c   e e − +   c+1 c+1  x (s ) = − c + √   √ 2 1−c2 s −2c 1−c2 s   e +2 e +  c+1 c+1 c+1 √ − c   (e2 1−c s − 1)   + c   y (s ) =   −2c √1−c2 s 2 2√1−c2 s   +2 + e e c+1 c+1 c+1 ❍❛② √ √  1−c2 s − 2e 1−c2 s + c ce   √ x (s) = − √ 2 1−c s − 2ce 1−c2 s + √ √ √e (e 1−c2 s − e− 1−c2 s )  − c y (s) =  √ √ 2 e 1−c s − 2c + e− 1−c s ✭✷✳✶✼✮ ✭✷✳✶✽✮ ✭✷✳✶✾✮ ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✶✾✮✱ ❝❤ó♥❣ t❛ ❝â √ e √1−c s −c − cs 1−c2 √ √ 2 e 1−c s + e− 1−c s − 2c   x(s) = arctan  y (s) = ln ❚r÷í♥❣ ❤đ♣ ❜✿ ✭✷✳✷✵✮ √ 1−c2 s √ b e 1−c s −1 ❤❛② tan ξ = +1 ❚➼♥❤ t♦→♥ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ ✶✱ ❝❤ó♥❣ t❛ ✤↕t ✤÷đ❝ √ tan ξ − b = e 1−c s tan ξ + b e √ e √1−c s +c 1−c2 √ √ 2 e 1−c s + e− 1−c s   x(s) = −2 arctan − cs  y (s) = ln + 2c ✭✷✳✷✶✮ ◆❤➟♥ ①➨t ✷✳✷✳✶✳ ✶✳ ❚❛ ✤➦t✿ g1 (c, s) = arctan ✸✺ e √ 1−c2 s √ −c − c2 − cs √ g2 (c, s) = −2 arctan √ g1 (c, s) = ln e g2 (c, s) = ln e 1−c2 s √ 1−c2 s e 1−c2 s √ +c − c2 √ + e− + e− 1−c2 s √ 1−c2 s − cs − 2c + 2c ❑❤✐ ✤â✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ g1 (c, s) = −g2 (−c, s), g1 (c, s) = g2 (−c, s) ❉♦ ✤â✱ ✈➳t ❝õ❛ ❤❛✐ ✤÷í♥❣ ❝♦♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✵✮ ✈➔ ✭✷✳✷✶✮ ❧➔ ✤è✐ ①ù♥❣ q✉❛ trö❝ Ox ✷✳ ❑❤✐ c = 0✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✵✮ ✈➔ ✭✷✳✷✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✤÷í♥❣ ●r✐♠ ✲ ❘❡❛♣❡r✳ x(s) = ±2 arctan(es ) ✭✷✳✷✷✮ s −s y (s) = ln(e + e ) ✷✳✷✳✷✳✹✳ ❚r÷í♥❣ ❤đ♣ c = ◆➳✉ tỗ t s0 s cos[2(s0)] = t (s) = ξ(s0) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✾✮✳ ❑❤✐ ✤â✱ α ❧➔ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ❤♦➦❝ trị♥❣ ✈ỵ✐ trư❝ Ox✳ ◆➳✉ cos[2ξ(s0)] = ✈ỵ✐ ♠å✐ s t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮ trð t❤➔♥❤ tan2 ξ −ξ = = + cos2 s + tan ξ ❉♦ ✤â✱ ❚ø ✤â✱ ❝❤ó♥❣ t❛ s✉② r❛ ✤÷đ❝ d(cot ξ ) = ds cot ξ ) = s ❑❤✐ ✤â✱ t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮ ❝❤ó♥❣ t❛ ❝â   x (s) = − s + s2  y (s) = 2s + s2 ✭✷✳✷✸✮ ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✷✸✮✱ ❝❤ó♥❣ t❛ ✤÷đ❝ x(s) = arctan s − s y (s) = ln(1 + s2 ) ✸✻ ✭✷✳✷✹✮ ❈❤ó♥❣ t❛ t❤➜② r➡♥❣ ❤❛✐ ✤÷í♥❣ ❝♦♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✺✮ ✈➔ ✭✷✳✷✹✮ ❝â ❝ị♥❣ ✈➳t ♥❤÷♥❣ ♥❣÷đ❝ ♥❤❛✉✳ ✷✳✷✳✷✳✺✳ ❚r÷í♥❣ ❤đ♣ c > ỵ tữỡ tỹ trữớ ủ c < −1✱ ❝❤ó♥❣ t❛ ❝â √ tan ξ = −a tan c2 − ✭✷✳✷✺✮ s ❉♦ ✤â✱ √ √  2−1 c c2 −  2  s d tan s − a tan   2  −   √ √ x (s ) = √   c2 − c2 − c2 −  2  + a tan s + tan s  2 √ √ c2 − c2 −   tan s d tan s    2 −4   √ √ y (s) =   2−1 c + c c2 −    s + tan2 s + a2 tan2 2 ▲➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✷✻✮ ✈➔ t❤❛② a = c−1 c+1 ✭✷✳✷✻✮ ✈➔♦ ❦➳t q✉↔✱ ❝❤ó♥❣ t❛ ✤÷đ❝  √  c − c2 −   tan s − cs x ( s ) = arctan   c+1      √  2−1 c s +1   tan      √ y (s) = − ln      c+1  c −   tan2 s +1  c−1 ❚ø ❝→❝ ❦➳t q✉↔ tr ú t t ữủ ỵ s ✷✳✷✳✹✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ (R2; ey )✱ ❝→❝ ✤÷í♥❣ ❝â f s ợ ữớ s ♠ët ♣❤➨♣ tà♥❤ t✐➳♥✳ ✶✳ ▼ët ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ ❜➡♥❣ ❤♦➦❝ ❧➔ ♠ët ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ợ trử Oy ữớ rr ữủ ❜ð✐ t❤❛♠ sè x(s) = arctan(es ) y (s) = ln(es + e−s ) ✸✼ ,s ∈ R ✭✷✳✷✼✮ ✷✳ ▼ët ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ ✤à♥❤ ❜ð✐ t❤❛♠ sè ❤♦➦❝ ❧➔ ♠ët ✤÷í♥❣ t❤➥♥❣ ❤♦➦❝ ❧➔ ✤÷đ❝ ①→❝ kf < √    x(s) = arctan e 1−c2 s √ −c − c2 − cs √ √   y (s) = ln e 1−c2 s + e− 1−c2 s − 2c ,s ∈ R ✸✳ ▼ët ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ ±1 ❤♦➦❝ ❧➔ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ trư❝ Ox ❤♦➦❝ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ t❤❛♠ sè x(s) = arctan s − s y (s) = ln(1 + s2 ) ,s ∈ R ✹✳ ▼ët ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ kf > ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ t❤❛♠ sè  √  c − c2 −   tan s x ( s ) = ± arctan   c+1      √  2−1 π π c ,√ tan2 s +  , s ∈ −√   c −1 c −1     √ y ( s ) = − ln       c−1 c2 −   tan s +1  c+1 ❍➺ q✉↔ ✷✳✷✳✸✳ ✶✳ ❍å ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❤ë✐ tö ✈➲ ♠ët ✤✐➸♠ trá♥ ❦❤✐ c ❞➛♥ r❛ ✈ỉ ❝ị♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû α(c) ❧➔ ♠ët ✤÷í♥❣ t❤❛♠ √ sè ❤â❛ ✤ë ❞➔✐ ❝✉♥❣✱ ❝â f ✲✤ë ❝♦♥❣ ❧➔ ♠ët ❤➡♥❣ sè c > 1✳ ❚❛ ✤➦t β (c) = c2 − 1α(c)✳ ❑❤✐ ✤â✱ ✤ë ❝♦♥❣ ❝õ❛ β ❜➡♥❣ √ k (α(c))✱ ð ✤â k (α(c)) ❧➔ ✤ë ❝♦♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ α(c)✳ ❚❤❡♦ c −1 ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮✱ k(α(c) = x + c✳ ❉♦ ✤â✱ 1 √ k (α(c)) = √ c2 − c2 − ❉➵ t❤➜② r➡♥❣✱ lim √ c→∞ c2 −1 √ −c cos( c2 − 1s) − √ +c c + cos( c2 − 1s) k (α(c)) = ❉♦ ✤â✱ ❤å ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❤ë✐ tö ✈➲ ♠ët ✤✐➸♠ trá♥ ❦❤✐ c ❞➛♥ r❛ ✈ỉ ❝ị♥❣✳ ✷✳ ❈→❝ ✤÷í♥❣ ❝â ✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ (R2; ey ) ❦❤æ♥❣ t❤➸ ❧➔ ❜✐➯♥ ❝õ❛ ♠ët ♠✐➲♥ ✤➥♥❣ ❝❤✉✳ ✷✳✷✳✷✳✻✳ ❍➻♥❤ ✈➩ ❝→❝ ✤÷í♥❣ ❝â f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ey ✸✽ ❍➻♥❤ ✷✳✶✿ k < −1 ❍➻♥❤ ✷✳✷✿ k = ±1 ❍➻♥❤ ✷✳✸✿ k ∈ (−1, 0) ❍➻♥❤ ✷✳✹✿ k = ❍➻♥❤ ✷✳✺✿ k ∈ (0, 1) ❍➻♥❤ ✷✳✻✿ k > ✸✾ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ❚ê♥❣ ❤đ♣✱ ❦❤→✐ q✉→t ✤÷đ❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ✤÷í♥❣ t❤❛♠ sè✱ ❝→❝ t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✤÷í♥❣ t❤❛♠ sè tr♦♥❣ R2 ✈➔ R3 ã r ỵ ố tr ổ ã ỵ ố ổ ❝á♥ ✤ó♥❣ ❤♦➔♥ t♦➔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➟t ✤ë t t ã ự ữủ ỵ ố tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤ ✤ó♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➔♠ ♠➟t ✤ë ❧➔ ❤➔♠ ❤➡♥❣✳ • r ợ ữớ õ f tr t ợ t ey ã ✣➲ t➔✐ ❦❤â❛ ❧✉➟♥ ❝â t❤➸ ♣❤→t tr✐➸♥ t❤❡♦ ❝→❝ ữợ s tr ỵ ố õ ỏ ú tr ổ ợ t ã ✣÷❛ r❛ ✤÷đ❝ ♠ët ♣❤➙♥ ❧♦↕✐ tr✐➺t ✤➸ ❝→❝ ✤÷í♥❣ f ✲✤ë ❝♦♥❣ ❤➡♥❣ tr➯♥ ♠➦t ♣❤➥♥❣ ✈ỵ✐ ♠➟t ✤ë ❧♦❣✲t✉②➳♥ t➼♥❤✳ • ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍✐❡✉ ❉✳ ❚✱ ❍♦❛♥❣ ◆✳ ▼✳✱ ▼ët sè ❦➳t q ữớ tr t ợ t t✉②➳♥ t➼♥❤✱ ❚↕♣ ❝❤➼ ❦❤♦❛ ❤å❝✱ ✣❍ ❍✉➳✱ ❚➟♣ ✼✹✱ ❙è ✺✱ ✭✷✵✶✷✮✱ ✺✸✲✻✹✳ ❬✷❪ ◆❛♠ ❚✳ ▲✳✱ ▼ët sè t t ữớ t tr ổ ợ ♠➟t ✤ë✱ ▲✉➟♥ →♥ t✐➯♥ s➽ t♦→♥ ❤å❝✱ ✭✷✵✶✺✮✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❇❛♥❝❤♦❢❢✱ ❚✳✱ ▲♦✈❡tt✱ ❙✳✱ ❉✐❢❢❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ❈❘❈ Pr❡ss✱ ❇♦❝❛ ❘❛t♦♥✱ ❋▲✱ ✷✵✶✻✳ ①✈✐✰✹✶✹ ♣♣✳ ■❙❇◆✿ ✾✼✽✲✶✲✹✽✷✷✲✹✼✸✹✲ ✸✳ ❬✹❪ ❇✐♥❤✳ ◆✳ ❉✱ ◆❛♠✳ ❚✳ ▲✳ ✭✷✵✶✸✮✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ❣❡♦❞❡s✐❝ ❝✉r✈❡s ✐♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❞❡♥s✐t②✱ ❊❛st ✲ ❲❡st ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✺ ✭✷✮✱ ✶✼✵✲✶✽✶✳ ❬✺❪ ❈❛r♠♦ ▼✳ P✳ ❉ ✭✶✾✼✻✮✱ ❉✐❢❢❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✱ Pr❡♥t✐❝❡✲ ❍❛❧❧✱ ❊♥❣❧❡✇♦♦❞ ❈❧✐❢❢s✱ ◆❏✳ ❬✻❪ ❈❛rr♦❧❧ ❈✳✱ ❏❛❝♦❜ ❆✳✱ ◗✉✐♥♥ ❈✳ ❛♥❞ ❲❛❧t❡rs ❘✳ ✭✷✵✵✽✮ ✱ ❚❤❡ ■s♦♣❡r✐♠❡tr✐❝ Pr♦❜❧❡♠ ♦♥ P❧❛♥❡s ✇✐t❤ ❉❡♥s✐t②✱ ❇✉❧❧✳ ❆✉str❛❧✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✽✱ ✶✼✼✲✶✾✼✳ ❬✼❪ ❈♦r✇✐♥ ■✳✱ ❍♦❢❢♠❛♥ ◆✳✱ ❍✉r❞❡r ❙✳✱ ❙❡s✉♠ ❱✳ ❛♥❞ ❳✉ ❨✳✱ ❉✐❢❢❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ♦❢ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❞❡♥s✐t②✱ ❘♦s❡✲❍✉❧♠❛♥ ❯♥❞✳ ▼❛t❤✳ ❏✳✱ ✼ ✭✶✮ ✭✷✵✵✻✮✳ ❬✽❪ ❍✐❡✉ ❉✳ ❚ ✲ ◆❛♠ ❉✳ ▲✱ ✭✷✵✶✹✮✱ ❚❤❡ ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❝♦♥st❛♥t ✇❡✐❣❤t❡❞ ❝✉r✈❛✲ t✉r❡ ❝✉r✈❡s ✐♥ t❤❡ ♣❧❛♥❡ ✇✐t❤ ❛ ❧♦❣✲❧✐♥❡❛r ❞❡♥s✐t②✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ♦♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆♥❛❧②s✐s✱ ✶✸✱ ✶✻✹✶✲✶✻✺✷✳ ❬✾❪ ❏✐❛♥ ❍✳✱ ❏✉ ❍✳✱ ▲✐✉ ❨✳ ❛♥❞ ❙✉♥ ❲✳ ✭✷✵✶✵✮✱ ❚r❛✈❡❧✐♥❣ ❢r♦♥ts ♦❢ ❝✉r✈❡ ❢❧♦✇ ✇✐t❤ ❡①t❡r♥❛❧ ❢♦r❝❡ ❢✐❡❧❞s✱ ❈♦♠♠✉♥✳ P✉r❡ ❆♣♣❧✳ ❆♥❛❧✳✱ ✾ ✭✹✮✱ ✾✼✺✲✾✽✻✳ ❬✶✵❪ ▼♦r❣❛♥ ❋✳ ✭✷✵✵✺✮✱ ▼❛♥✐❢♦❧❞s ✇✐t❤ ❞❡♥s✐t②✱ ◆♦t✐❝❡s ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✺✷✱ ✽✺✸✲✽✺✽✳ ❬✶✶❪ ▼♦r❣❛♥ ❋✳ ✭✷✵✵✽✮✱ ●❡♦♠❡tr✐❝ ▼❡❛s✉r❡ ❚❤❡♦r②✱ ❆ ❇❡❣✐♥♥❡r✬s ●✉✐❞❡✱ ❋♦✉rt❤ ❡❞✐t✐♦♥✱ ❆❝❛❞❡♠✐❝ Pr❡ss✳ ❬✶✷❪ ◆❛♠ ❚✳ ▲✳ ✭✷✵✶✹✮✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ❝✉r✈❡s ✐♥ ♣❧❛♥❡s ✇✐t❤ ❛ ❧♦❣✲❧✐♥❡❛r ❞❡♥s✐t②✱ ❙♦✉t❤❡❛st ❆s✐❛♥ ❇✉❧❧❡t✐♥ ♦❢ ▼❛t❤♠❡t✐❝s✱ ❛❝❝❡♣t❡❞✳ ❬✶✸❪ ❖✬◆❡✐❧❧ ❇✳✱ ❊❧❡♠❡♥t❛r② ❉✐❢❢❡r❡♥t✐❛❧ ●❡♦♠❡tr②✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ▲♦♥❞♦♥✲◆❡✇ ❨♦r❦✱ ✶✾✻✻✳ ❬✶✹❪ ❚♦♣♦♥♦❣♦✈✱ ❱✳ ❆✳✱ ❉✐❢❢❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ♦❢ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✳ ❆ ❝♦♥❝✐s❡ ❣✉✐❞❡✱ ❲✐t❤ t❤❡ ❡❞✐t♦r✐❛❧ ❛ss✐st❛♥❝❡ ♦❢ ❱❧❛❞✐♠✐r ❨✳ ❘♦✈❡♥s❦✐✳ ❇✐r❦❤❛✉s❡r ❇♦st♦♥✱ ✹✶ ■♥❝✳✱ ❇♦st♦♥✱ ▼❆✱ ✷✵✵✻✳ ①✐✈✰✷✵✻ ♣♣✳ ■❙❇◆✿ ✾✼✽✲✵✲✽✶✼✻✲✹✸✽✹✲✸❀ ✵✲✽✶✼✻✲✹✸✽✹✲✷✳ ❲❡❜s✐t❡ ❬✶✺❪ ✇✇✇✳❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✳ ❬✶✻❪ ✇✇✇✳♠❛t❤✇♦r❧❞✳✇♦❧❢r❛♠✳❝♦♠✳ ✹✷