surface integralssurface integralssurface integrals, Lecture notes of Mathematics

Download surface integralssurface integralssurface integrals and more Lecture notes Mathematics in PDF only on Docsity! We are booking for parametricequatirepreseuh.mgsurfaces . { " lui D= ? Y (u , u ) = ? 2 ( u , u ) = ? ✓(u , u) = xlu , u) i + ylu , r)j + 2 ( u , r) K ☒ : a Sphere of radius a { ✗ = as inclus y = a soin ¢ sino 2 = acos ¢ ✓(¢ , G) =asinocoso-i-asi.no/sin0-j-acosQk j'Ë of radius a Ex ' : a Sphere-centered at P " (% , yo /27 { ✗ = as inclus + x. y = a soin ¢ sino + Yo z = acos ¢ + Zo EI : P= ( l , 2 , 3) À= < l , o , o > À =U , I , o> Find the paramétrisation of the plane passing through P and containing the rectus I and Î . ① T'✗ v7 should be normal to that plane = | i j k pI O O I I I O I = o . i - j -1k Thus the plaines equation is - ( y - 1) + (2-3)=0 a paramétrisation is { " = u y = v 2 = ✓ + 2 This is crucial for describingtengentp-a.ws ✓(u , u) = alu , v) il y/u , u)j +Ku , u) K O O rv = 2£ Luo , vo) i +2¥ Luo , volj-F-ulu.is ) te Rende : tangent curve of grid curve ! ru = JI Luo , vo) i + 0£ (no , volj-F-ulu.is ) k & If ru ✗ vu -1-0 S is smooth the tangent plane is the plane containing ru , ru and ru x ru is a normal vector to the tangentplane . EI : Sphere of radius a { x-asinocosoy-asinosi.no D= §z = a ces ¢ rolxvo =p ' ' j le acosdcoso-acosdsmo-asi.no//-asinOsin0asin0/casO-0 = a2sin20coso-i-is.in?cfsinO-j-a2sin0coscfk/rqXro-/--a4sin4#s0-- l-a4sinkfces.RO/--/a2srn0l Als) = sino / 0 0 = 21T a Z GU-surfaaareao-fgrap.hu { ✗ = xy = y 2 = f- (x, y) ⇒ rx = i + ¥ , le ry = j + ¥yk ⇒ rx ✗ ry =/ i j k l'o ? ˧ / = -¥ i -¥ : + le ⇒ lrxxryt-F.IE?--yiT like we saws in section 15.5 Torus is surface of Revolution of C . z " + lyt# Iabout 2-avis ✗= , /tofu) Cos v • 2)Et 5in y = " + coussin OÇUE 21T 0 EVE21T C ; { y = 2 + "S " is a parameter: Z = Sin u of e . satin OE -0421T ① 2 does not change value ② fake the angle v into accounts = (2-icosuuscosvx4j-2-cos.ie Y = (2 + (osu) sinv So a paramétrisation Ë÷÷ "" y= (2 + osu) Sinn