Differential Geometry Homework (22 вариант. ДЗ1. ФН.)

Äîìàøíåå çàäàíèå 1Êðèâûå è ïîâåðõíîñòè â ïðîñòðàíñòâåÑòóäåíò:Ãðóïïà:Âàðèàíò:Ïðåïîäàâàòåëü:1234567~r = {5u cos v, 4u sin v, u2 };γ : u = v; P1 (0, 0, 0), P2 (−5π, 0, π 2 ). äåêàðòîâûõ êîîðäèíàòàõ:x = 5u cos v;y = 4u sin v;z = u2 .x2y2+= 2z.25/28Ýëëèïòè÷åñêèé ïàðàáîëîèä.1.

Íàéòè îñîáûå òî÷êè ïàðàìåòðèçîâàííîé ïîâåðõíîñòè S : ~r = ~r(u, v), u, v ∈ R.Ñîñòàâèòü óðàâíåíèå êàñàòåëüíîé ïëîñêîñòè ê ïîâåðõíîñòè â òî÷êàõ P1 , P2 .Íàéä¼ì îñîáûå òî÷êè ñ ïîìîùüþ ìàòðèöû ßêîáè. Íàéä¼ì ~ru , ~rv .~ru=~rv=∂~r∂u∂~r∂vJ=={5 cos v, 4 sin v, 2u};={−5u sin v, 4u cos v, 0}.5 cos v−5u sin v4 sin v4u cos v12u0T.Òàê êàê sin v è cos v íå ìîãóò îäíîâðìåííî îáðàòèòüñÿ â íîëü, òî ýëåìåíòû âòîðîé ñòðîêè òðàíñïîíèðîâàííîé ìàòðèöû ßêîáè îáðàòÿòñÿ â íîëüòîëüêî ïðè íóëåâûõ çíà÷åíèÿõ u. Çíà÷èò ðàíã ìàòðèöû ßêîáè áóäåò ðàâåíäâóì ïðè u 6= 0. Òîãäà îñîáûå òî÷êè: (0, v), ∀v ∈ R.Îáëàñòü îïðåäåëåíèÿ: D = {u, v|u > 0, v ∈ (0, 2π)}. S = ϕ(D).Íàéä¼ì çíà÷åíèÿ ïàðàìåòðîâ u è v , ñîîòâåòñòâóþùèå òî÷êàì P1 è P2 :0 = 5u1 cos v1 ;0 = 2u1 sin v1 ;0 = u21 .(u1 = 0;v1 = 0.Òàê êàê u = 0, òî òî÷êà P1 îñîáàÿ.−5π = 5u2 cos v2 ;−0 = 4u2 sin v2 ; 2π = u22 .(u2 = π;v2 = π.(u3 = −π;v3 = 0.Ïàðà (u3 , v3 ) íå ðàññìàòðèâàåòñÿ, òàê îíà íå ëåæèò â îáëàñòè îïðåäåëåíèÿ ïîâåðõíîñòè.

Óðàâíåíèå êàñàòåëüíîé ïëîñêîñòè ê ãëàäêîé ïîâåðõíîñòèS : ~r = ~r(u, v) â òî÷êå P (u0 , v0 ) èìååò âèä:~r = ~r(u0 , v0 ) + u~ru (u0 , v0 ) + v~rv (u0 , v0 ),u, v ∈ R.Ïîäñòàâèâ â ýòî óðàâíåíèå âìåñòî u0 , v0 ïàðó u2 , v2 , ïîëó÷èì óðàâíåíèåêàñàòåëüíîé ïëîñêîñòè â òî÷êå P2 :~r = −5π − 5u, −4πv, π 2 + 2πu .2. Èññëåäîâàòü çàâèñèìîñòü âèäà ïîâåðõíîñòè îò îáëàñòè èçìåíåíèÿ ïàðàìåòðîâ (u, v). Ñîñòàâèòü óðàâíåíèÿ êîîðäèíàòíûõ ëèíèé. Ïîñòðîèòü ïîâåðõíîñòü è êîîðäèíàòíóþ ñåòü íà íåé (ñ èñïîëüçîâàíèåì ñèñòåìû êîìïüþòåðíîé àëãåáðû Mathematica).Óðàâíåíèÿ êîîðäèíàòíûõ ëèíèé:1. Ïåðâîå ñåìåéñòâî ~r = {5u0 cos v, 4u0 sin v, u20 }, u0 = const, v ∈ (0, 2π) Ýëëèïñû áåç òî÷êè.2. Âòîðîå ñåìåéñòâî ~r = {5u cos v0 , 4u sin v0 , u2 }, v0 = const, u > 0 Âåòâèïàðàáîë.2u ∈ [0, 2], v ∈ [0, 2π] :u ∈ [1, 3], v ∈ [0, 2π] :3u ∈ [0, 3], v ∈ [0, 2π] :π 11π]:u ∈ [0, 3], v ∈ [ ,2 643. Âû÷èñëèòü ïåðâóþ êâàäðàòè÷íóþ ôîðìó ïîâåðõíîñòè.

Âû÷èñëèòüóãîë ìåæäó êðèâûìè u = v 2 è u = v â òî÷êå èõ ïåðåñå÷åíèÿ.Äëÿ ñîñòàâëåíèÿ ïåðâîé êâàäðàòè÷íîé ôîðìû âû÷èñëèì:E = (~ru , ~ru ) = 25 cos2 v + 16 sin2 v + 4u2 ;F = (~ru , ~rv ) = −9u cos v sin v;G = (~rv , ~rv ) = 25u2 sin2 v + 16u2 cos2 v.Ïåðâàÿ êâàäðàòè÷íàÿ ôîðìà:I = Edu2 + 2F dudv + Gdv 2 == (25 cos2 v+16 sin2 v+4u2 )du2 −18u cos v sin v dudv+(25u2 sin2 v+16u2 cos2 )dv 2 .Ìàòðèöà ïåðâîé êâàäðàòè÷íîé ôîðìû: E F25 cos2 v + 16 sin2 v + 4u2G==F G−9u cos v sin v−9u cos v sin v.25u2 sin2 v + 16u2 cos2 vÊðèâûå γ1 : u = v 2 è γ2 : u = v ïàðàìåòðèçóåì ñëåäóþùèì îáðàçîì:(u = t2 ;γ1 :v = t.(u = t;γ2 :v = t.

2tξ1 = γ̇1 =.1 1ξ2 = γ̇2 =.1Òî÷êàìè ïåðåñå÷åíèÿ êðèâûõ γ1 è γ2 áóäóò òî÷êè t = 0 è t = 1. Íî òàêêàê ïðè t = 0 ïîëó÷èì u = 0, òî áåð¼ì t = 1. Êîñèíóñ óãëà ìåæäó êðèâûìèíà ïîâåðõíîñòè âû÷èñëèì ïî ôîðìóëå:cos (\γ1 , γ2 )=cos (\γ1 , γ2 )t=1=(\γ1 , γ2 )≈t=1ξ1T Gξ2p.ξ1T Gξ1 ξ2T Gξ2139 + 9 cos 2 − 27 sin 2p.2 3/2 (79 + 9 cos 2 − 12 sin 2) (45 − 9 sin 2)p21.78◦ .54. Âû÷èñëèòü âòîðóþ êâàäðàòè÷íóþ ôîðìó ïîâåðõíîñòè.

Îïðåäåëèòüòèïû òî÷åê ïîâåðõíîñòè.Äëÿ ñîñòàâëåíèÿ âòîðîé êâàäðàòè÷íîé ôîðìû âû÷èñëèì:8u2 cos v− √400u2 +82u4 −18u4 cos 2v~ru × ~rv10u2 sin v√−~n ==2 +82u4 −18u4 cos 2v  .400uk~ru × ~rv k2220u cos v+20u sin v√~ruu=~rvv=~ruv=∂ 2~r∂u2∂ 2~r∂v 2∂ 2~r∂u∂vL = (~ruu , ~n) = √400u2 +82u4 −18u4 cos 2v={0, 0, 2};={−5 sin v, 4 cos v, 0};={−5u cos v, −4u sin v, 0}.400u240u;+ 82u4 − 18u4 cos 2vM = (~ruv , ~n) = 0;N = (~rvv , ~n) = √40u3.400u2 + 82u4 − 18u4 cos 2vÂòîðàÿ êâàäðàòè÷íàÿ ôîðìà:II = Ldu2 + 2M dudv + N dv 2 ==√40u40u3du2 + √dv 2 .400u2 + 82u4 − 18u4 cos 2v400u2 + 82u4 − 18u4 cos 2vÌàòðèöà âòîðîé êâàäðàòè÷íîé ôîðìû:40L M1B=·=√M F0400u2 + 82u4 − 18u4 cos 2v0.u2Òàê êàê √400u2 +82u40> 0, ∀u ∈ R, òî ñëåäîâàòåëüíî det B > 0, ∀u > 0, ∀v ∈ (0, 2π).4 −18u4 cos 2vÀ çíà÷èò âñå òî÷êè ïîâåðõíîñòè ýëëèïòè÷åñêèå.65.

Íàéòè ãëàâíûå íàïðàâëåíèÿ è ãëàâíûå êðèâèçíû â òî÷êàõ P1 è P2 .Âû÷èñëèòü ñðåäíþþ è ãàóññîâó êðèâèçíó ïîâåðõíîñòè â ýòèõ òî÷êàõ.P1 îñîáàÿ òî÷êà. P2 (−5π, 0, π 2 ), u = π, v = π.GBdet |B − kG|=P225 + 4π 20√=40400+64π 2= (√P2;!0;2√ 40π400+64π 20P2016π 24040π 2− k(25 + 4π 2 ))( √− k(16π 2 )).400 + 64π 2400 + 64π 2Ãëàâíûå êðèâèçíû â òî÷êå P2 :k1 =105√, k2 = √.(25 + 4π 2 ) 25 + 4π 28 25 + 4π 2B − kG=00=)√− 85(9+4π25+4π 20P2 ,k1P2 ,k2;210π (9+4π )√(25+4π 2 ) 25+4π 22B − kG!0200!.Ãëàâíûå íàïðàâëåíèÿ â òî÷êå P2 :B − kGP2 ,k1ξ1 = 0̄ ⇔00!02210π (9+4π )√(25+4π 2 ) 25+4π 22B − kGP2 ,k2 ξ2 = 0̄ ⇔)√− 85(9+4π25+4π 2000!ξ11ξ12ξ21ξ22==Ãàóññîâà è ñðåäíÿÿ êðèâèçíû â òî÷êå P2 :K=H=k1 k2k1 + k22=254(25+4π 2 )2 ;=5(41+4π 2 )√.16(25+4π 2 ) 25+4π 270000⇒ ξ1 =⇒ ξ2 =1001;.6. Ñîñòàâèòü óðàâíåíèÿ ëèíèé êðèâèçíû, àñèìïòîòè÷åñêèõ ëèíèé è ãåîäåçè÷åñêèõ äëÿ ðàññìàòðèâàåìîé ïîâåðõíîñòè.

Ïðèâåñòè ïðèìåðû ðåøåíèÿýòèõ óðàâíåíèé.Óðàâíåíèÿ àñèìïòîòè÷åñêèõ ëèíèé:Lu̇2 + 2M u̇v̇ + N v̇ 2 = 0, ãäåu̇ è v̇ ýòî(ïðîèçâîäíûå ïî t ñîîòâåòñòâóþùèõ ôóíêöèé àñèìïòîòè÷åñêèõu = u(t), à L, M, N ýòî ñîîòâåòñòâóþùèå êîýôôèöèåíòûëèíèé γ :v = v(t)âòîðîé êâàäðàòè÷íîé ôîðìû.Òàê êàê âñå òî÷êè ýëëèïòè÷åñêèå, òî àñèìïòîòè÷åñêèõ ëèíèé íåòó.7. Âû÷èñëèòü êðèâèçíó êðèâîé γ : u = v â òî÷êàõ P1 è P2 .  îäíîé èçòî÷åê ïîñòðîèòü ðåïåð Ôðåíå.~r = {5t cos t, 4t sin t, t2 };~r˙ = {5 cos t − 5t sin t, 4 sin t + 4t cos t, 2t};~r¨ = {−5t cos t − 10 sin t, 8 cos t − 4t sin t, 2}.Êðèâèçíà:k=k P1=k P2=k~r˙ × ~r¨k;k~r˙ k3√2 1725 ;q22 85+ 466π+25π 45.5(5+4π 2 )3/2Íàéäåì ðåïåð Ôðåíå â òî÷êå P1 = (0, 0, 0) : t = 0:Åäèíè÷íûé êàñàòåëüíûé âåêòîð:Âåêòîð áèíîðìàëè:Âåêòîð ãëàâíîé íîðìàëè:~τ P1~r˙ =k~r˙ k ~r˙ × ~r¨ β~ P1 =k~r˙ × ~r¨k P1~~ν P = β × ~τ P1{1, 0, 0} ;=P11==140, − √ , √;1717410, √ , √.1717~.Ïîðÿäîê â ðåïåðå Ôðåíå: ~τ , ~ν , βÂñå âû÷èñëåíèÿ ïðîèçâåäåíû â ñèñòåìå êîìïüþòåðíîé àëãåáðû Wolfram Mathematica.8.