TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* PHẠM NGỌC MAI ÁNH XẠ GAUSS TRONG HỆ TỌA ĐỘ ĐỊA PHƯƠNG KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Hình học HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* PHẠM NGỌC MAI ÁNH XẠ GAUSS TRONG HỆ TỌA ĐỘ ĐỊA PHƯƠNG KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chun ngành: Hình học Người hướng dẫn khoa học TS TRẦN VĂN NGHỊ HÀ NỘI – 2018 ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ▲í✐ ❝❛♠ ✤♦❛♥ ✷ ▲í✐ ♥â✐ ✤➛✉ ✸ ✶ ⑩◆❍ ❳❸ ●❆❯❙❙ ✺ ✶✳✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ❈→❝ ♠➦t ❝❤➼♥❤ q✉② ✈➔ t↕♦ ↔♥❤ ❝õ❛ ❝→❝ ❣✐→ trà ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✷ ❍➔♠ ❦❤↔ ✈✐ tr➯♥ ♠➦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✸ ▼➦t ♣❤➥♥❣ t✐➳♣ ①ó❝ ❝õ❛ ♠➦t ❝♦♥❣ ✈➔ ✈✐ ♣❤➙♥ ❝õ❛ →♥❤ ①↕ ✶✳✶✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✷ ⑩♥❤ ①↕ ●❛✉ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸ ❈æ♥❣ t❤ù❝ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✹ ✣ë ❝♦♥❣ ●❛✉ss ✷✺ ✶✳✷ ❉↕♥❣ ❝ì ❜↔♥ t❤ù ♥❤➜t ✈➔ ❞✐➺♥ t➼❝❤ ✶✶ ⑩♥❤ ①↕ ●❛✉ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ⑩◆❍ ❳❸ ●❆❯❙❙ ❚❘❖◆● ❍➏ ❚➴❆ ✣❐ ✣➚❆ P❍×❒◆● ✐ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✸✸ ✷✳✶ ⑩♥❤ ①↕ ●❛✉ss tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❑➳t ❧✉➟♥ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✶ ✐✐ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲í✐ ❝↔♠ ì♥ ◗✉❛ ✹ ♥➠♠ ❤å❝ t➟♣ ✈➔ r➧♥ ❧✉②➺♥ t↕✐ ❣✐↔♥❣ ✤÷í♥❣ ✣↕✐ ❤å❝ ✱ ✤÷đ❝ sü ❞➻✉ ❞➢t ✈➔ ❞↕② ❞é ❝õ❛ ❝→❝ ❚❤➛② ❝æ ❣✐→♦✱ ❡♠ ✤➣ t✐➳♣ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❤ú✉ ➼❝❤ ✈➔ q✉❛♥ trå♥❣✳ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔② ❝❤➼♥❤ ❧➔ t❤➔♥❤ q✉↔ ❝õ❛ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ r➧♥ ❧✉②➺♥ ✤â✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❣✐ó♣ ❡♠ ❝â ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❦❤♦❛ ❤å❝ ❝ơ♥❣ ♥❤÷ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♠➻♥❤✳ õ ữủ t ữợ sỹ ữợ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❚❤➛② ❣✐→♦ ❚❙✳❚r➛♥ ❱➠♥ ◆❣❤à✳ ❊♠ ①✐♥ ♣❤➨♣ ❣û✐ ✤➳♥ t❤➛② ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t✳ ❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❣✐ó♣ ✤ï✱ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥ ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ✈ø❛ q✉❛✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ P❤↕♠ ◆❣å❝ ▼❛✐ ✶ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲í✐ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✧⑩♥❤ ①↕ ●❛✉ss tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ tø sü ♥é ❧ü❝ t t ũ ợ sỹ ữợ ❞➝♥ ❝õ❛ ❚❤➛② ❣✐→♦ ❚❙✳❚r➛♥ ❱➠♥ ◆❣❤à✳ ❚r♦♥❣ ❜➔✐ ❧✉➟♥ ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✭♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✮✳ ❱➻ ✈➟② ❡♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❡♠✱ ❦❤æ♥❣ trò♥❣ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ P❤↕♠ ◆❣å❝ ▼❛✐ ✷ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲í✐ õ ỵ t ✈✐ ♣❤➙♥ ❧➔ ♥❣➔♥❤ ❤➻♥❤ ❤å❝ ❝❤✉②➯♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ✈➔ ✤à♥❤ ❧÷đ♥❣ ❝õ❛ ❝→❝ ❤➻♥❤ ❤➻♥❤ ❤å❝ ♥❤í ❝→❝ ❝ỉ♥❣ ❝ư ❝õ❛ ❣✐↔✐ t➼❝❤ ♠➔ trü❝ t✐➳♣ ❧➔ ♣❤➨♣ t➼♥❤ ✈✐ t➼❝❤ ♣❤➙♥✳ ❈→❝ ♠➦t ✤è✐ ❝❤✐➲✉ ♠ët✱ ♠➦t tr♦♥❣ E3 , s✐➯✉ ♠➦t tr♦♥❣ En ❧➔ ❝→❝ ✤è✐ t÷đ♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ❍➻♥❤ ❤å❝ ✈✐ ♣❤➙♥✳ ✣➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ✤è✐ t÷đ♥❣ tr➯♥✱ ♥❣÷í✐ t❛ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ●❛✉ss✳ ⑩♥❤ ①↕ ●❛✉ss ❧➔ ♠ët ❝æ♥❣ ❝ư ❤ú✉ ➼❝❤ ✤➸ ✤÷❛ r❛ ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➲ ✤ë ❝♦♥❣ ●❛✉ss✱ ✤ë ❝♦♥❣ tr✉♥❣ ❜➻♥❤✱ ✤ë ❝♦♥❣ ❝❤➼♥❤✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❝→❝ ✤è✐ tữủ õ tr ữủ sỹ ữợ t ữợ qt t ✤➸ tr➻♥❤ ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✤↕✐ ❤å❝ ♥❣➔♥❤ ❙÷ ♣❤↕♠ ❚♦→♥ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉✳ ❛✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❧➔ →♥❤ ①↕ ●❛✉ss ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❝❤ó♥❣✳ ❜✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ỵ tt ss ♥❣❤✐➯♥ ❝ù✉✳ ✸ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❧➔♠ rã ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➲ →♥❤ ①↕ ●❛✉ss ✈ỵ✐ ❝→❝ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✿ ♠➦t ❝❤➼♥❤ q✉②✱❤➔♠ ❦❤↔ ✈✐ tr➯♥ ♠➦t✱ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ♥❤➜t ✈➔ ❞✐➺♥ t➼❝❤ tr➯♥ ♠ët ♠➦t✱ t ữợ ss ✳ ✳❚➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ ●❛✉ss tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ P❤➙♥ t➼❝❤ ✈➔ tê♥❣ ❤đ♣ ❦✐➳♥ t❤ù❝✳ ✺✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥✳ õ ỗ ữỡ ữỡ ●❛✉ss ❈❤÷ì♥❣ ✷✿ ⑩♥❤ ①↕ ●❛✉ss tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣ ✹ ❈❤÷ì♥❣ ✶ ⑩◆❍ ❳❸ ●❆❯❙❙ ❚r♦♥❣ ❝❤÷ì♥❣ ✶ ♥➔② ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ữ s ởt số ỡ ❝õ❛ ♠➦t ❝❤➼♥❤ q✉② tr♦♥❣ R3 , ♠æ t↔ ♠ët sè t✐➯✉ ❝❤➼ ❤ú✉ ➼❝❤ ✤➸ ✤÷❛ r❛ ♠ët t➟♣ ❝♦♥ ♥❤➜t ✤à♥❤ ❝õ❛ R3 ❧➔ ♠ët ♠➦t ❝❤➼♥❤ q✉②✳ • ✣à♥❤ ♥❣❤➽❛ ♠ët ❤➔♠ tr➯♥ ♠➦t ❝❤➼♥❤ q✉② ❧➔ ❤➔♠ ❦❤↔ ✈✐✱ t❛ s➩ t❤➜② r➡♥❣ ❦❤→✐ ♥✐➺♠ ✈✐ ♣❤➙♥ tr♦♥❣ R2 ❝â t❤➸ ✤÷đ❝ ♠ð rë♥❣ ✤➳♥ ❝→❝ ❤➔♠ ♥❤÷ ✈➟②✳ ❉♦ ✤â ♠➦t ❝❤➼♥❤ q✉② tr♦♥❣ R3 ❝❤♦ t❛ ♠ët sü t❤✐➳t ❧➟♣ tü ♥❤✐➯♥ tø ❝→❝ ♣❤➨♣ t➼♥❤ ❤❛✐ ❝❤✐➲✉✳ • ❉↕♥❣ ❝ì ❜↔♥ t❤ù ♥❤➜t ữ ổ tỹ ỷ ỵ ❤ä✐ ✤à♥❤ ❧÷đ♥❣ ✭✤ë ❞➔✐ ❝õ❛ ✤÷í♥❣ ❝♦♥❣✱ ❞✐➺♥ t➼❝❤ ❝õ❛ t❤✐➳t ❞✐➺♥ ✳ ✳ ✳ ✮ tr➯♥ ♠ët ♠➦t q t ữợ ①↕ ●❛✉ss ✈➔ ✤ë ❝♦♥❣ ●❛✉ss ✺ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✶✳✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳✶ ❈→❝ ♠➦t ❝❤➼♥❤ q✉② ✈➔ t↕♦ ↔♥❤ ❝õ❛ ❝→❝ ❣✐→ trà ❝❤➼♥❤ q✉② ▼➦t ❝❤➼♥❤ q✉✐ tr♦♥❣ R3 ❝â t❤➸ ❤➻♥❤ ❞✉♥❣ ♥❤÷ s❛✉✿ ▲➜② ♠ët sè ♠↔♥❤ ♠➦t ♣❤➥♥❣ ❜✐➳♥ ❞↕♥❣ ❝❤ó♥❣ ✈➔ ✏❞→♥✑ ❧↕✐ s❛♦ ❝❤♦ ❤➻♥❤ ♥❤➟♥ ✤÷đ❝ ❦❤ỉ♥❣ ❝â ❝→❝ ✤✐➸♠ ♥❤å♥✱ ❦❤ỉ♥❣ ❝â ❝→❝ ❝↕♥❤ ❤♦➦❝ ❦❤æ♥❣ ❝â t➼♥❤ tü ❝➢t ✤➸ t↕✐ ♠é✐ ✤✐➸♠ ❝â t❤➸ ♥â✐ ✤➳♥ ♠➦t ♣❤➥♥❣ t✐➳♣ ①ó❝ ❝õ❛ ♠➦t✳ ❈→❝ ♠➦t ❝ơ♥❣ s➩ ✤÷đ❝ ❣✐↔ t❤✐➳t ✤õ trì♥ ✤➸ ❝â t❤➸ ♠ð rë♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ơ♥❣ ♥❤÷ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❣✐↔✐ t➼❝❤ ❧➯♥ ❝❤ó♥❣✳ ✣à♥❤ ♥❣❤➽❛ s❛✉ ✤➙② t❤ä❛ ♠➣♥ ❝→❝ ②➯✉ ❝➛✉ tr➯♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❝❤➼♥❤ q✉✐ ♥➳✉ ∀p ∈ S X : U → V ∩ S, ✈ỵ✐ ▼ët t➟♣ ❤đ♣ ❝♦♥ tỗ t U S R3 V R3 ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠➦t ❝õ❛ ❧➔ ♠ët t➟♣ ❝♦♥ ♠ð ❝õ❛ p ✈➔ →♥❤ ①↕ R2 ✱ t❤ä❛ ♠➣♥ ✸ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ⑩♥❤ ①↕ X ❧➔ ❦❤↔ ✈✐✱ ❝â ♥❣❤➽❛ ❧➔ X(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ U ✷✳ ⑩♥❤ ①↕ ✈ỵ✐ X x, y, z ❧➔ ❝→❝ ❤➔♠ ❝â r ởt ỗ ổ tứ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ✶✱ ♥➯♥ ①↕ ♥❣÷đ❝ X ✈➔♦ V S X tử ởt ỗ ♣❤æ✐ ❝â ♥❣❤➽❛ ❧➔ X −1 : V ∩ S → U ❝❤➳ ❝õ❛ ♠ët →♥❤ ①↕ ❧✐➯♥ tö❝ t➟♣ ♠ð ❝❤ù❛ U ❧✐➯♥ tö❝✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ F : W ⊂ R3 → R3 X X −1 ❝â →♥❤ ❧➔ ❤↕♥ ①→❝ ✤à♥❤ tr➯♥ ♠ët V ∩ S ✸✳ ✭❚➼♥❤ ❝❤➼♥❤ q✉✐✮ ❱ỵ✐ ♠å✐ ✤ì♥ →♥❤✳ ⑩♥❤ ①↕ X q ∈ U, ✈✐ ♣❤➙♥ DXq : R2 → R2 ❧➔ ♠ët ✤÷đ❝ ❣å✐ ❧➔ ♠ët t❤❛♠ sè ❤â❛ ✭✤à❛ ♣❤÷ì♥❣✮ ✻ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤♦ ❜ð✐✿ d = X(u, v) − X(0, 0), N (p) ❱➻ X(u, v) ❧➔ ❦❤↔ ✈✐✱ t❛ ❝â ❝æ♥❣ t❤ù❝ ❚❛②❧♦r ✿ X(u, v) = X(0, 0) + Xu u + Xvv + (Xuu u2 + 2Xuv uv + Xvv v ) + R, tr♦♥❣ ✤â✱ ❝→❝ ✤↕♦ ❤➔♠ ✤÷đ❝ ❧➜② t↕✐ (0, 0) ✈➔ ♣❤➛♥ ❞÷ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ R =0 (u,v)→(0,0) u2 + v lim ❙✉② r❛ d = X(u, v) − X(0, 0), N (p) = Xuu , N (p) u2 + Xuv , N (p) uv + Xvv , N (p)v + R 1 = eu2 + 2f uv + gv + R = + IIp(w) + R, 2 w = Xu u + Xv v, R = R, N (p) , tr♦♥❣ ✤â ❈❤♦ ♠ët ✤✐➸♠ ❡❧❧✐♣t✐❝ ❣➛♥ p, IIp(w) R |w|2 lim w→0 = ❝â ❞➜✉ ❝è ✤à♥❤✳ ❉♦ ✤â✱ ✈ỵ✐ ∀(u, v) ✤õ p, d ❝â ❝ò♥❣ ợ IIp(w); tự (u, v) ữ t❤✉ë❝ ❝ò♥❣ ♠ët ♣❤➼❛ ✈ỵ✐ T p(S) ❈❤♦ ♠ët ✤✐➸♠ ❤②♣❡r❜♦❧✐❝ (u, v) ✈➔ (u, v) ✭tr♦♥❣ ✤â✱ ♥❤❛✉ ❝õ❛ s❛♦ ❝❤♦ p, tr♦♥❣ ♠é✐ ❧➙♥ ❝➟♥ ❝õ❛ IIp(w/|w|) w = Xuu + Xvv); IIp(w/|w|) p tỗ t ✤✐➸♠ ❝â ❝ò♥❣ ❞➜✉ ❞÷ì♥❣ ❝→❝ ✤✐➸♠ ♥❤÷ ✈➟② t❤✉ë❝ ❤❛✐ ♣❤➼❛ ❦❤→❝ T p(S) ❈❤♦ ♠ët ✤✐➸♠ ❡❧❧✐♣t✐❝ ❣➛♥ ✈➔ p, IIp (w) ❝â ❞➜✉ ❝è ✤à♥❤✳ ❉♦ ✤â✱ ✈ỵ✐ ∀(u, v) ✤õ p, d ❝â ❝ò♥❣ ❞➜✉ ✈ỵ✐ IIp (w); tù❝ ❧➔✱ ♠å✐ (u, v) ♥❤÷ ✈➟② t❤✉ë❝ ❝ò♥❣ ✸✽ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ♠ët ♣❤➼❛ ✈ỵ✐ Tp S ✈➔ (u, v) ❞➜✉ ❞÷ì♥❣ ✭tr♦♥❣ ✤â✱ ♣❤➼❛ ❦❤→❝ ♥❤❛✉ ❝õ❛ s❛♦ ❝❤♦ IIp (w/|w|) w = Xu u + X v v ❀ ✈➔ IIp (w/|w|) ❝â ❝ò♥❣ ❝→❝ ✤✐➸♠ ♥❤÷ ✈➟② t❤✉ë❝ Tp (S) ú ỵ t õ t →♣ ❞ö♥❣ ❝❤♦ ❧➙♥ ❝➟♥ ❝õ❛ ♠ët ✤✐➸♠ ♣❛r❛❜♦❧✐❝ ❤♦➦❝ ♠ët ✤✐➸♠ ♣❤➥♥❣✳ ❑❤✐ ✤â ❝→❝ ✤✐➸♠ ♣❛r❛❜♦❧✐❝ ✈➔ ✤✐➸♠ ♣❤➥♥❣ ❝õ❛ ♠➦t ♥➡♠ tr➯♥ ♠ët ♣❤➼❛ ❝õ❛ ♠➦t ♣❤➥♥❣ t ú õ t õ ởt ữớ ợ ♠➦t ♣❤➥♥❣ ♥➔②✳ ❱➼ ❞ö ✷✳✷✳✶✳ ▼➦t ✧▼♦♥❦❡②s❛❞❞❧❡✧ ✭❳❡♠ ❤➻♥❤ ✷✳✷✮ ✤÷đ❝ ❝❤♦ ❜ð✐ x = u, ❚➼♥❤ trü❝ t✐➳♣✱ t↕✐ ✤✐➸♠ e = f = g = 0; z = u3 − 3v u y = v, (0, 0) ❞♦ ✤â✱ ✤✐➸♠ ❝→❝ ❤➺ sè ❝õ❛ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ ❧➔ (0, 0) ❧➔ ♠ët ✤✐➸♠ ♣❤➥♥❣✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❜➜t ❦➻ ❝õ❛ ✤✐➸♠ ♥➔②✱ ❝â ❝→❝ ✤✐➸♠ ♥➡♠ tr➯♥ ❝↔ ❤❛✐ ♣❤➼❛ ❝õ❛ ♠➦t ♣❤➥♥❣ t✐➳♣ ①ó❝ ❝õ❛ ♥â✳ ❱➼ ❞ư ✷✳✷✳✷✳ ❳➨t ♠➦t trá♥ ①♦❛② t❤✉ ✤÷đ❝ ❜ð✐ ✤÷í♥❣ ❝♦♥❣ z = y , < z < 1, ợ ữớ z=1 ✭❳❡♠ ❤➻♥❤ ✷✳✸✮✳ ❘ã r➔♥❣ ❝→❝ ✤✐➸♠ ✤÷đ❝ s✐♥❤ ❜ð✐ sü q✉❛② q✉❛♥❤ ❣è❝ ❖ ❧➔ ❝→❝ ✤✐➸♠ ♣❛r❛❜♦❧✐❝✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ ♥❣➢♥ ❣å♥ r➡♥❣ ❝→❝ ✤÷í♥❣ s♦♥❣ s♦♥❣ ✈➔ ❝→❝ ✤÷í♥❣ ❦✐♥❤ t✉②➳♥ ❝õ❛ ♠ët ♠➦t trá♥ ①♦❛② ❧➔ ❝→❝ ✤÷í♥❣ ❝❤➼♥❤❀ ✭❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝â ❞↕♥❣ y = x3 ✮ ❝â ✤ë ❝♦♥❣ ❜➡♥❣ ✈➔ ✤÷í♥❣ s♦♥❣ s♦♥❣ ❧➔ ♠ët t✐➳t ❞✐➺♥ ✈✉ỉ♥❣ õ ú ỵ r tr t ởt r ữ tỗ t↕✐ ❝→❝ ✤✐➸♠ ♥➡♠ tr♦♥❣ ❝↔ ❤❛✐ ♣❤➼❛ ❝õ❛ ♠➦t ♣❤➥♥❣ t✐➳♣ ①ó❝✳ ❇✐➸✉ t❤ù❝ ❝õ❛ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣ r➜t ❤ú✉ ➼❝❤ ❝❤♦ ✸✾ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍➻♥❤ ✷✳✷ ❍➻♥❤ ✷✳✸ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ t✐➺♠ ❝➟♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ ❝❤➼♥❤✳ ✣➛✉ t✐➯♥✱ t❛ ①➨t ❝→❝ ♣❤÷ì♥❣ t✐➺♠ ❝➟♥✳ ▲➜② X(u, v) ❧➔ ♠ët t❤❛♠ sè ❤â❛ t↕✐ e(u, v) = e; f (u, v) = f, ✈➔ g(u, v) = g p ∈ S, ✈ỵ✐ X(0, 0) = p, ✈➔ ❧➜② ❧➔ ❝→❝ ❤➺ sè ❝õ❛ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ tr♦♥❣ t❤❛♠ sè ❤â❛ ♥➔②✳ ❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣✱ ♠ët ✤÷í♥❣ ❝♦♥❣ ❝❤➼♥❤ q✉② ❤➺ tå❛ ✤ë ❧➙♥ ❝➟♥ ❝õ❛ ♥➳✉ t❤❛♠ sè ❤â❛ ❜➜t ❦➻ II(α (t)) = 0, ✈ỵ✐ X C ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ t✐➺♠ ❝➟♥ ♥➳✉ ✈➔ ❝❤➾ α(t) = X(u(t), v(t)), ∀t ∈ I, ❧✐➯♥ t❤ỉ♥❣ ✤÷đ❝ tr♦♥❣ ✈ỵ✐ t ∈ I, ❝õ❛ C t❛ ❝â tù❝ ❧➔✱ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ e(u )2 + 2f u v + g(v )2 = 0, t∈I ✭✷✳✶✮ P❤÷ì♥❣ tr➻♥❤ ✷✳✶ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ t✐➺♠ ❝➟♥✳ ❚ø ♣❤÷ì♥❣ tr➻♥❤ ✷✳✶ t❛ ❝â ❦➳t ❧✉➟♥ s❛✉✿ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ ♠ët t❤❛♠ ❤â❛ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ ♠ët ✤✐➸♠ ❤②♣❡r❜♦❧✐❝ ✭❣✐↔ sû✱ −f < 0✮ ❧➔ ❝→❝ ✤÷í♥❣ ❝♦♥❣ t✐➺♠ ❝➟♥ ❚ù❝ ❧➔✱ ♥➳✉ ❝↔ ❤❛✐ ✤÷í♥❣ ❝♦♥❣ e = g = u = const, v = v(t) ✈➔ u = u(t), u = const t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✷✳✶✱ t❛ ❝â e = g = ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✤✐➲✉ ✹✵ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❦✐➺♥ ♥➔② t❤ä❛ ♠➣♥✳ ❇➙② ❣✐í t❛ ①➨t ❝→❝ ♣❤÷ì♥❣ ❝❤➼♥❤✳ ▼ët ✤÷í♥❣ ❝♦♥❣ ❝❤➼♥❤ q✉② ❝➟♥ ❝õ❛ r C ❧✐➯♥ t❤ỉ♥❣ ✤÷đ❝ tr♦♥❣ ❤➺ tå❛ ✤ë ❧➙♥ ữớ ợ t số ❤â❛ ❜➜t ❦➻ α(t) = X(u(t), v(t)) ❝õ❛ C, t ∈ I, t❛ ❝â d(N (α (t))) = λ(t)α (t) ❈→❝ ❤➔♠ u (t), v (t) t❤ä❛ ♠➣♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ f E − eG gF − f G u + v =λu EG − F EG − F eF − f E f F − gE u + v =λv EG − F EG − F ❑❤û λ tr♦♥❣ ❤➺ tr➯♥✱ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❝❤➼♥❤ ❦❤ó❝ (f E − eF )(u )2 + (gE − eG)u v + (gF − f G)(v )2 = ✭✷✳✷✮ ❤❛② (v )2 −u v (u )2 E F G e f g = ❘ã r➔♥❣ ❝→❝ ♣❤÷ì♥❣ ❝❤➼♥❤ ❧➔ trỹ ợ õ tứ ữỡ ❝➛♥ ✈➔ ✤õ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ tå❛ ✤ë ❝õ❛ ♠ët t❤❛♠ sè ❤â❛ ❧➔ ❝→❝ ✤÷í♥❣ ❝❤➼♥❤ ❦❤ó❝ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ♠ët ✤✐➸♠ ❦❤æ♥❣ rè♥ ❧➔ F = f = 0✳ tr➻♥❤ ✷✳✷ t❛ ❝â ❦❤➥♥❣ ✤à♥❤ s❛✉✳ ❱➼ ❞ö ✷✳✷✳✸✳ ✭▼➦t trá♥ ①♦❛②✮ ❳➨t ♠ët ♠➦t trá♥ ①♦❛② ✤÷đ❝ t❤❛♠ sè ✹✶ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❤â❛ ❜ð✐ X(u, v) = (ϕ(v) cos u, ϕ(v) sin u, ψ(v)), < u < 2π, a < v < b, ϕ(v) = ❈→❝ ❤➺ sè ❝õ❛ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ♥❤➜t ✤÷đ❝ ❝❤♦ ❜ð✐ E = ϕ2 , F = 0, G = (ϕ )2 + (ψ )2 ✣➸ t❤✉➟♥ t✐➺♥✱ ❣✐↔ sû r➡♥❣ ✤÷í♥❣ ❝♦♥❣ q✉❛② ✤÷đ❝ t❤❛♠ sè ❤â❛ ❜ð✐ ✤ë ❞➔✐✱ tù❝ ❧➔ (ϕ )2 + (ψ )2 = G = ❉➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ ❝→❝ ❤➺ sè ❝õ❛ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ ♥❤÷ s❛✉✿ −ϕ sin u ϕ cos u −ϕ cos u (Xu , Xv , Xuv ) e= √ = ϕ cos u EG − F EG − F ϕ sin u ϕ sin u ψ = −ϕψ, f =0 g =ψϕ −ψ ϕ ✭v = const✮ ❱➻ F = f = 0, t❛ ❦➳t ❧✉➟♥ r➡♥❣ ❝→❝ ✤÷í♥❣ s♦♥❣ s♦♥❣ ✈➔ ❝→❝ ❦✐♥❤ t✉②➳♥ ✭u = const✮ ❝õ❛ ♠ët ♠➦t trá♥ ①♦❛② ❧➔ ❝→❝ ✤÷í♥❣ ❝❤➼♥❤ ❦❤ó❝ ❝õ❛ ♠ët ♠➦t ♥❤÷ ✈➟②✿ ❱➻✿ eg − f ψ (ψ ϕ − ψ ϕ ) K=− = EG − F ϕ ✈➔ ϕ ❤♦➦❝ ❧✉ỉ♥ ❞÷ì♥❣✱ ✤✐➲✉ ✤â ❝❤ù♥❣ tä r➡♥❣ ❝→❝ ✤✐➸♠ ♣❛r❛❜♦❧✐❝ ✤➣ ❝❤♦ ψ = q ữớ t ú ợ ữớ s ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ trư❝ ϕψ − ψ ϕ = ✭✤ë ❝♦♥❣ ❝õ❛ ✤÷í♥❣ s✐♥❤ ❜➡♥❣ ❦❤ỉ♥❣✮✳ ▼ët ✤✐➸♠ t❤ä❛ ♠➣♥ ❝↔ ❤❛✐ ✤✐➲✉ ❦✐➺♥ tr➯♥ ❧➔ ♠ët ✤✐➸♠ ♣❤➥♥❣✱ ✈➻ ✈➟② ✹✷ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ e = f = g = t❛ ❝â ❚ø (ϕ )2 + (ψ )2 = −ψ ϕ ❧➜② ✈✐ ♣❤➙♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❛ ✤÷đ❝ ϕψ = ❉♦ ✤â ψ (ϕ ψ − ψ ϕ ) (ψ )2 ϕ + (ϕ )2 ϕ ϕ K=− =− =− ϕ ϕ ϕ ✭✷✳✸✮ P❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❧➔ ❜✐➸✉ t❤ù❝ ❝❤♦ ✤ë ❝♦♥❣ ●❛✉ss ❝õ❛ ♠➦t trá♥ ①♦❛②✳ P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â t❤➸ ✤÷đ❝ →♣ ❞ư♥❣ ✤➸ t➼♥❤ ✤ë ❝♦♥❣ ●❛✉ss ❦❤æ♥❣ ✤ê✐ ❝õ❛ ♠➦t trá♥ ①♦❛②✳ ✣➸ t➼♥❤ ❝→❝ ✤ë ❝♦♥❣ ❝❤➼♥❤✱ ✤➛✉ t✐➯♥✱ t❛ ✤÷❛ r❛ ♥❤➟♥ ①➨t tê♥❣ q✉→t ◆➳✉ ♠ët t❤❛♠ sè ❤â❛ ❝õ❛ ♠ët ♠➦t ❝❤➼♥❤ q✉② ❝â E = f = 0, t❤➻ ❝→❝ ✤ë ❝♦♥❣ ❝❤➼♥❤ ✤÷đ❝ ❝❤♦ ❜ð Ee ✈➔ Gg ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❝→❝ ✤ë s❛✉✿ ❝♦♥❣ ●❛✉ss ✈➔ ❞ë ❝♦♥❣ tr✉♥❣ ❜➻♥❤ ✤÷đ❝ ❝❤♦ ❜ð✐ K= ❱➻ K ❧➔ t➼❝❤ ✈➔ 2H eg eG + gE ,h = EG EG ❧➔ tê♥❣ ❝õ❛ ❝→❝ ✤ë ❝♦♥❣ ❝❤➼♥❤✱ ♥➯♥ ❝→❝ ✤ë ❝♦♥❣ ❝❤➼♥❤ ❝õ❛ ♠ët ♠➦t trá♥ ①♦❛② ✤÷đ❝ ❝❤♦ ❜ð✐ e ψϕ ψ =− =− E ϕ ϕ g =ψϕ − ψ ϕ ; G ✭✷✳✹✮ ❑❤✐ ✤â✱ ✤ë ❝♦♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♠ët ♠➦t ♥❤÷ ✈➟② ❧➔ H= ❱➼ ❞ư ✷✳✷✳✹✳ z = h(x, y), −ψ + ϕ(ψ ϕ − ψ ϕ ) = ϕ ✭✷✳✺✮ ▼ët t ữủ ữ ỗ t ởt ✈✐ tr♦♥❣ ✤â (x, y) t❤✉ë❝ ♠ët t➟♣ ♠ð ✹✸ U ⊂ R2 P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚❤❛♠ sè ❤â❛ ♠➦t tr➯♥ ❜ð✐ X(u, v) = (u, v, h(u, v)), (u, v) ∈ U, tr♦♥❣ ✤â u = x, v = y ❚❛ ❝â ✿ Xu =(1, 0, hu ), Xv = (0, 1, hv ), Xuu =(0, 0, huu ), Xuv = (0, 0, huv ), Xvv =(0, 0, hvv ) ❉♦ ✤â N (x, y) = (−hx , −hy , 1) (1 + h2x + h2y ) ❧➔ ♠ët tr÷í♥❣ ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à tr➯♥ ❝→❝ ♠➦t✱ ✈➔ ❝→❝ ❤➺ số ỡ tự tr ỵ ♥➔② ✤÷đ❝ ❝❤♦ ❜ð✐ e= f= g= hxx (1 + h2x + h2y ) hxy (1 + h2x + h2y ) hyy (1 + h2x + h2y ) ❉ü❛ ✈➔♦ ❝→❝ ❜✐➸✉ t❤ù❝ tr➯♥✱ t❛ ❝â t❤➸ t➼♥❤ ✤÷đ❝ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❦❤→❝ ♠ët ❝→❝❤ ❞➵ ❞➔♥❣✳ ❱➼ ❞ö✱ tø ✭✶✳✺✮ t❛ t➼♥❤ ✤ë ❝♦♥❣ ●❛✉ss ✈➔ ✤ë ❝♦♥❣ ✹✹ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♠➦t✿ hxx hxy − h2xy K= ; (1 + h2x + h2y )2 (1 + h2x )hyy − 2hx hy hxy + (1 + h2y )hxx H= (1 + h2x + h2y ) ◆❤➟♥ ①➨t✿ ❈❤♦ ♠ët ✤✐➸♠ p t❤✉ë❝ ♠ët ♠➦t S ❈❤å♥ trö❝ tå❛ ✤ë ❝õ❛ R3 s❛♦ ❝❤♦ ❣è❝ ❝õ❛ S t↕✐ O p ❝õ❛ tå❛ ✤ë ❧➔ t↕✐ p ✈➔ trö❝ z ❙✉② r❛✱ ♠ët p tr ữợ z = h(x, y), (x, y) ⊂ U ⊂ R2 , t➟♣ ♠ð ✈➔ h ❧➔ ❤➔♠ ❦❤↔ ✈✐ ✈ỵ✐ ❞å❝ t❤❡♦ ♣❤→♣ t✉②➳♥ ❞÷ì♥❣ S ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❤➻♥❤ ✷✳✹✱ tr♦♥❣ ✤â U ❧➔ ♠ët h(0, 0) = p, hx (0, 0) = 0, hy (0, 0) = ❍➻♥❤ ✷✳✹✿ ▼é✐ ✤✐➸♠ ❝õ❛ S ❝â ♠ët ❧➙♥ ❝➟♥ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ❞↕♥❣ z = h(x, y) ❉↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝õ❛ S t↕✐ p →♣ ❞ư♥❣ ✈ỵ✐ ✈❡❝tì ✹✺ (x, y) ∈ R2 , tr♦♥❣ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✈➼ ❞ö ♥➔② ❝â ❞↕♥❣ hxx (0, 0)x2 + 2hxy (0, 0)xy + hxy (0, 0)y ❉↕♥❣ ❜➟❝ ❤❛✐ tr➯♥ ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤÷ ❧➔ ❞↕♥❣ ❍❡ss✐❛♥ ❝õ❛ ❤ t↕✐ ✤✐➸♠ (0, 0) ❉♦ ✤â✱ ❞↕♥❣ ❍❡ss✐❛♥ ❝õ❛ ❤ t↕✐ (0, 0) ❧➔ ❞↕♥❣ ❝ì ❜↔♥ t❤ù ❤❛✐ ❝õ❛ S t↕✐ p ⑩♣ ❞ư♥❣ ❝→❝ ♥❤➟♥ ①➨t tr➯♥ t❛ s➩ ✤÷❛ r❛ ữủ ởt t ỗ ữ s >0 ọ tũ ỵ s C = {(x, y) ∈ Tp (S); h(x, y) = ε} ❧➔ ✤÷í♥❣ ❝♦♥❣ ❝❤➼♥❤ q✉② ✭t❛ ❝â t❤➸ t❤❛② ✤ê✐ ữợ t õ > p ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ✤✐➸♠ ♣❤➥♥❣✱ t❤➻ ✤÷í♥❣ ❝♦♥❣ ✧❣➛♥ ❣✐è♥❣✧ ✈ỵ✐ t↕✐ p ✭❍➻♥❤ ✷✳✺✮✳ ●✐↔ sû ❝→❝ trư❝ ❝→❝ ♣❤÷ì♥❣ ❝❤➼♥❤✱ trư❝ z ❞å❝ t❤❡♦ ♣❤÷ì♥❣ ❝õ❛ ✤ë ỹ ỗ õ S C f = hxy (0, 0) = x ✈➔ y ❞å❝ t❤❡♦ ✈➔ e = hxx (0, 0), E g k2 (p) = = hyy (0, 0) G k1 (p) = ❑❤❛✐ tr✐➸♥ h(x, y) tr♦♥❣ ❝æ♥❣ t❤ù❝ ❚❛②❧♦r ♠ð rë♥❣ t↕✐ ✤✐➸♠ ✹✻ (0, 0), ✈➔ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝♦✐ hx (0, 0) = = hy (0, 0), t❛ ❝â h(x, y) = (hxx (0, 0)x2 + 2hxy (0, 0)xy + hyy (0, 0)y ) + R = (k1 x2 + k2 y ) + R, tr♦♥❣ ✤â R = (u,v)→(0,0) x2 + y lim ❉♦ ✤â✱ ✤ë ❝♦♥❣ C ✤÷đ❝ ❝❤♦ ❜ð✐ k1 x2 + k2 y + 2R = 2ε, k1 x2 + k2 y = 2ε ❧➔ √ ♠ët ①➜♣ ①➾ ❜➟❝ ♥❤➜t ❝õ❛ C ❇✐➳♥ ✤ê✐ t÷ì♥❣ tü✱ t❛ ✤÷đ❝ x = x 2ε, y = √ y 2ε, t❛ ❝â ❱➻ ✈➟②✱ ♥➳✉ p ❧➔ ✤✐➸♠ ❦❤æ♥❣ ♣❤➥♥❣✱ t❛ ❝â t❤➸ ❝♦✐ k1 x2 − k2 y = ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ tr♦♥❣ ✤÷í♥❣ trá♥ s❛✉ k1 x2 + k2 y = 1, ❧➔ ♠ët ✤✐➸♠ ❦❤æ♥❣ ♣❤➥♥❣✱ t❤➻ ❣✐❛♦ ❝õ❛ ♠➦t ♣❤➥♥❣ s♦♥❣ s♦♥❣ ❝õ❛ Tp(S) ✈ỵ✐ S ✤â♥❣ ✈ỵ✐ p, tr♦♥❣ ♠ët ①➜♣ ①➾ ❜➟❝ ♥❤➜t✱ ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❣✐è♥❣ ợ ỗ t p ỗ ❉✉♣✐♥ t↕✐ ◆➳✉ p p, ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r➡♥❣✱ ♥➳✉ p ❧➔ ♠ët ✤✐➸♠ ♣❤➥♥❣✱ sü ❣✐↔✐ t❤➼❝❤ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣✳ ✹✼ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍➻♥❤ ✷✳✺ ◆❤➟♥ ①➨t ▲➜② S ✈➔ S t q ữợ p S, dϕp →♥❤ ①↕ ❦❤↔ ✈✐ ✈➔ ❣✐↔ sû ✈ỵ✐ ϕ:S→S ❧➔ ♠ët ❦❤æ♥❣ s✉② ❜✐➳♥✳ ❚❛ ♥â✐ r➡♥❣ ϕ t ữợ t p õ ♠ët ❝ì sð ❞÷ì♥❣ {w1, w2} tr♦♥❣ Tp (S), ❦❤✐ ✤â {dϕp (w1 ), dϕp (w2 )} ❦❤ỉ♥❣ ❧➔ ❝ì sð ❞÷ì♥❣ t❤➻ t❛ ♥â✐ ϕ ❧➔ p →♥❤ ①↕ ữủ ữợ t ú ỵ S t ❝➛✉ ✤ì♥ ✈à S ✤➲✉ ①➨t tr♦♥❣ R3 õ ởt ữợ N s tr dNp S s ợ ởt ữợ ổ s ❱ỵ✐ ♠ët ❝ì sð N tr♦♥❣ {w1 , w2 } S ▲➜② p ∈ S tr♦♥❣ Tp (S) dNp (w1 ) ∧ dNp (w2 ) = det(dNp )(w1 ∧ w2 ) = K(w1 ∧ w2 ), →♥❤ ①↕ ss N ữủ ữợ t s t ữợ t pS ởt ữợ ọ tr S Tp (S) K(p) < p∈S ♥➳✉ K(p) > ✈➔ ✤ê✐ ❱➻ ✈➟②✱ t❛ ❝â ❦➳t ❧✉➟♥ s❛✉✿ ◆➳✉ ự ởt ữợ ữớ q p; t❤➻ ↔♥❤ N ❝õ❛ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ♥➔② s➩ ũ ữợ ữủ ữợ ợ ởt ữợ ố ❧÷đt ♣❤ư t❤✉ë❝ ✈➔♦ p ❧➔ ♠ët ✤✐➸♠ ❡❧❧✐♣t✐❝ ❤❛② ♠ët ✤✐➸♠ ❤②♣❡r❜♦❧✐❝ ✭❍➻♥❤ ✶✳✺✮ ✹✽ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍➻♥❤ ✷✳✻✿ ⑩♥❤ ①↕ ●❛✉ss t ữợ t ởt t ữủ ữợ t ởt r t K(p) = ❧➔ ♠ët ❧➙♥ ❝➟♥ ❧✐➯♥ t❤ỉ♥❣ ✤÷đ❝ ❝õ❛ ✈➔ ❧➜② V p S ❧➔ ♠ët ✤✐➸♠ ❝õ❛ s❛♦ ❝❤♦ ✤ë ❝♦♥❣ ●❛✉ss p tr♦♥❣ ✤â K ❦❤æ♥❣ ✤ê✐ ❞➜✉✳ ❑❤✐ ✤â A , A→0 A K(p) = lim tr♦♥❣ ✤â A ❧➔ ♠ët t➼❝❤ ♣❤➙♥ ❝õ❛ ♠ët ♠✐➲♥ ♣❤➙♥ ❝õ❛ ↔♥❤ B ❜ð✐ →♥❤ ①↕ ●❛✉ss q✉❛ ♠ët ụ q q ú ỵ p Bn N : S → S 2, ♠➔ ❤ë✐ tư tỵ✐ ❝❤ù❛ ♠å✐ Bn B⊂V p, ❝❤ù❛ p, A ❧➔ t➼❝❤ ợ ữủ õ t t ❝➛✉ ♥➔♦ ✤õ ❧ỵ♥✳ ❙♦ s→♥❤ ❜✐➸✉ t❤ù❝ ❝õ❛ t➼♥❤ ❝❤➜t tr➯♥ ✈ỵ✐ ❜✐➸✉ t❤ù❝ ❝õ❛ ✤ë ❝♦♥❣ δ S→0 S k = lim ✹✾ P❤↕♠ ◆❣å❝ ▼❛✐ ❑❤â❛ ❧✉➟♥ tốt ởt ữớ t tỹ ợ ✤ë ❝♦♥❣ k C t↕✐ p, t❛ t❤➜② r➡♥❣ ✤ë ❝♦♥❣ ●❛✉ss K ❧➔ t÷ì♥❣ s ❧➔ ✤ë ❞➔✐ ❝õ❛ ❝→❝ ✤÷í♥❣ ♣❤➥♥❣ ❝❤♦ ❝→❝ ♠➦t ✭ð ✤➙②✱ ❝õ❛ ♠ët ✤♦↕♥ ♥❤ä ❝õ❛ C ❝❤ù❛ p, ✈➔ ❝õ❛ ✤÷í♥❣ t✐➳♣ t✉②➳♥✮✳ ✺✵ δ ❧➔ ✤ë ❞➔✐ ❝õ❛ ↔♥❤ tr♦♥❣ ❝❤➾ ỗ t r t ở ❝õ❛ ✤➲ t➔✐ ✧⑩♥❤ ①↕ ●❛✉ss tr♦♥❣ ❤➺ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣✧✳ ❚r♦♥❣ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔② ❡♠ ✤➣ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣✱ rã r➔♥❣ ✈➲ ❝→❝ ✤➦❝ tr✉♥❣ ✈➔ t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❝õ❛ →♥❤ ①↕ ●❛✉ss✳ ❑❤â❛ ❧✉➟♥ ✤➣ ✤↕t ✤÷đ❝ ♠ư❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ö ✤➲ r❛✳ ❚✉② ♥❤✐➯♥ ❞♦ ♥ë✐ ❞✉♥❣ ❝á♥ ❦❤→ ♠ỵ✐ ♠➫ ✈➔ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳✱ ♠ët ♣❤➛♥ ✈➻ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ rt ữủ ỳ ỵ õ õ qỵ ❜→✉ ❝õ❛ ❝→❝ ❚❤➛②✱ ❈æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ❝â t❤➸ ✤➛② ✤õ ✈➔ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼✳P✳❞♦ ❈❛r♠♦✱ ❉✐❢❢❡r❡♥t✐❛❧ ●❡♦♠❡tr② ♦❢ ❈✉r✈❡s ❛♥❞ ❙✉r❢❛❝❡s✱ Pr❡♥t✐❝❡✲❍❛❧❧ ✭✶✾✼✻✮✳ ❬✷❪ ❱ô ❚❤à ❚❤✉ ❍✐➲♥✱ ❍➻♥❤ ❤å❝ ✈✐ ♣❤➙♥ ❝õ❛ ❝→❝ ♠➦t ❝❤➼♥❤ q✉②✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣❍❙P ỵ →♥❤ ①↕ ●❛✉ss✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✤↕✐ ❤å❝✱ ❚r÷í♥❣ ✣❍❙P ❍➔ ◆ë✐ ✷✱✷✵✶✹ ✺✷ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* PHẠM NGỌC MAI ÁNH XẠ GAUSS TRONG HỆ TỌA ĐỘ ĐỊA PHƯƠNG KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Hình học Người hướng dẫn khoa