《数学分析》课程教学课件（讲稿，打印版）第11章 曲线积分和曲面积分（2/7）

­¡¡È ­¡È©½Â ­¡È©5Ÿ §11.2 1.­¡È© 11.2.1 ­¡¡È  S ´Ü1wëê­¡: ~r = ~r(u, v) = x(u, v)~i + y(u, v)~j + z(u, v)~k, (u, v) ∈ D, = x(u, v), y(u, v), z(u, v) äkëY ê, … ~r0 u × ~r0 v 6= 0 ùp D ´ëCþ (u, v) ¤3²¡¥‡k¡È«. ~r(u, v) Ò´‡²¡« D  R3 KN. 1/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ ^²1u Ouv ‹I¶†‚ u = ui, v = vj ©« D, Ù¥ ‡« Dij = {(u, v) : ui 6 u 6 ui+1, vj 6 v 6 vj+1} éA3 S þ Ò‡f­¡ Sij, §dü^ u ­‚ v = vj, v = vj+1 Úü^ v ­‚ u = ui, u = ui+1 Œ¤.  ∆ui = ui+1 − ui Ú ∆vj = vj+1 − vj Ñéž, Sij Œ±Cqw¤d ~r(ui+1, vj) − ~r(ui, vj) Ú ~r(ui, vj+1) − ~r(ui, vj) ܤ²1 o>/. O u v O x y z 2/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ Ϗ ~r(ui+1, vj) − ~r(ui, vj) = ~r0 u (ui, vj)∆ui + o(∆ui), ~r(ui, vj+1) − ~r(ui, vj) = ~r0 v (ui, vj)∆vj + o(∆vj), ¤±, Sij ¡È ∆Sij ≈ |~r0 u (ui, vj) × ~r0 v (ui, vj)|∆ui∆vj. u´­¡ S ¡È S = ZZ D |~r0 u (u, v) × ~r0 v (u, v)| dudv. 3/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ du (~r0 u × ~r0 v ) 2 = ~r02 u ~r02 v − (~r0 u · ~r0 v ) 2 , · E = ~r02 u = x 02 u + y 02 u + z 02 u , G = ~r02 v = x 02 v + y 02 v + z 02 v , F = ~r0 u · ~r0 v = x 0 ux 0 v + y 0 u y 0 v + z 0 u z 0 v , Ò­¡ S þ¡Èƒ dS = |~r0 u (u, v) × ~r0 v (u, v)| dudv = p EG − F2dudv, (11.1) Ú­¡ S ¡È„OŽúª S = ZZ D p EG − F2dudv. (11.2) 4/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ  F = ~r0 u · ~r0 v = 0 ž, `² u ­‚Ú v ­‚ƒ•þƒpR†, ¤± u ­‚Ú v ­‚3­¡þ?„ ‡ƒpR†­‚. ­¡ S ´²¡«ž, e S ëꐧ x = x(u, v), y = y(u, v), z = 0, (u, v) ∈ D. K |~r0 u × ~r0 v | =    ∂(x,y) ∂(u,v)    , ¡Èƒ dσ =    ∂(x,y) ∂(u,v)    dudv. 2g­È©¥²¡« S Ú²¡« D ƒmC†¥¡Èƒƒm 'X. 5/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ XJ­¡ S ´½Â3« D þ¼ê z = f(x, y) ‰Ñ, ùžŒò x, y wŠëCþ, ­¡ S ëꐧÒAO/ª ~r = ~r(x, y) = (x, y, f(x, y)) (x, y) ∈ D, Ï ~r0 x = (1, 0, f0 x ), ~r0 y = (0, 1, f0 y ). u´ E = 1 + f 02 x , G = 1 + f 02 y , F = f 0 x f 0 y . dd dS = q 1 + f 02 x + f 02 y dxdy 9 S = ZZ D q 1 + f 02 x + f 02 y dxdy. (11.3) XJ­¡§ x = g(y, z) ½ y = h(z, x), (J´aq. 6/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ ~ 1 ¦Œ» R ¥L¡È. ) ¥¡ S ëꐧ ~r : x = R sin θ cos ϕ, y = R sin θ sin ϕ, z = R cos θ, (0 6 θ 6 π, 0 6 ϕ 6 2π). OŽŒ ~r0 θ = (R cos θ cos ϕ, R cos θ sin ϕ, −R sin θ), ~r0 ϕ = (−R sin θ sin ϕ, R sin θ sin ϕ, 0). E = R2 , F = 0, G = R2 sin2 θ. ¤±¥¡¡È‡´ dS = R2 sin θdθdϕ. , ¥¡¡È S = Z π 0 dθ Z 2π 0 R2 sin θdϕ = 4πR2 . 7/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ ~ 2 ¦¥¡ x 2 + y 2 + z 2 = R2 Ρ x 2 + y 2 = Rx ¤e­¡¡ È. ) dué¡5, Œ¤¦­¡¡È´§31%S¡Èo. æ^¥¡‹Iž dS = R2 sin θdθdϕ. (½ëê θ, ϕ Cz« D, r¥ ¡ëꐧ\ѐ§ x 2 + y 2 = Rx Œk sin θ = cos ϕ, Ϗ31 %¥ 0 6 θ 6 π 2 , 0 6 ϕ 6 π 2 , ¤±ùüÜ­¡‚3¥¡‹Ie§  θ = π 2 − ϕ. Ïd D : 0 6 ϕ 6 π 2 , 0 6 θ 6 π 2 − ϕ, l Ž S = 4 ZZ D R2 sin θdθdϕ = 4R2 Z π 2 0 dϕ Z π 2 −ϕ 0 sin θ dθ = 4R2 Z π 2 0 (1 − sin ϕ)dϕ = 2R2 (π − 2). x y z 8/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ 11.2.2 1.­¡È© ­¡†áŸþ ­¡ S ´3 R3 ¥‡šþ!†á, ٟþ©Ù (¡ Ý) ëY¼ê ρ(x, y, z). ¦d†áŸþ. Ün{´kò†á S ©¤k‡¿©†á¡: S1, S2, · · · , Sn, z¡þŸþÝCq‡~ê ρ(ξi, ηi, ζi), ùp (ξi, ηi, ζi) ¡ Si þ:, ¤±¡ŸþCq ρ(ξi, ηi, ζi)∆Si , Ù¥ ∆Si ´¡ Si ¡ È. ò¤kùCqŠƒ\Ò´†áoŸþCqŠ: X n i=1 ρ(ξi, ηi, ζi)∆Si. ©©[ž, ù‡Úª4ŠATÒ´†áŸþ. 9/18 kJ Ik J I ˆ£ ¶ '4 òÑ

­¡¡È ­¡È©½Â ­¡È©5Ÿ ½Â 1  S ´Ük.1w­¡, f(x, y, z) ´½Â3 S þ¼ê. ^?¿©{r S ©¤ n ¬k¡È­¡ S1, S2, · · · , Sn, ù ­¡¬¡ ÈP ∆Si . 3z¬­¡ Si þ?: Pi(ξi, ηi, ζi), XJe4 lim λ→0 X n i=1 f(ξi, ηi, ζi)∆Si, ´‡kê, …† Pi(ξi, ηi, ζi) ÀJÃ', Ù¥ λ ´¤k¬­¡ Œ†», K¡¼ê f(x, y, z) 3­¡ S þŒÈ, 4ŠÒ´§È©Š, P ¤ ZZ S f(x, y, z)dS = lim λ→0 X n i=1 f(xi, yi, zi)∆Si, Ù¥ dS ¡­¡¡Èƒ. ZZ w, ~Š¼ê f = c 3k¡È σ(S) ­¡ S þ´ŒÈ, … S c dS = cσ(S). 10/18 kJ Ik J I ˆ£ ¶ '4 òÑ