Gauss úª Stokes úª ©/ªÈ© §11.5 Gauss ½nÚ Stokes ½n 11.5.1 Gauss ½n Green úªÑ ²¡¥þ| F~ = (P, Q) ÷´ L þ I L P dx + Qdy ¼ê ∂Q ∂x − ∂P ∂y 3 L SÜ D È©m'X, = I L P dx + Qdy = ZZ D ∂Q ∂x − ∂P ∂y dxdy. ùúª£ã ¼êÛÜ5N5méX, `² 3½^ eÛÜ/±û½N/. @okµ4¡þ1.¡È©¡¤¤n4«þ nÈ©kvkéXQ? 1/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© Bu* kl{üµ4¡ . V = [0, a] × [0, b] × [0, c] ´ R3 ¥N, S = ∂V ´µ4 ¡, §k8ý¡ S1, · · · , S6, { ©O ~n1, · · · , ~n6. F~ = (P, Q, R) ´ V þ1wþ|, K x y z O n n n n n n 1 2 3 4 5 6 a b c ZZ S1 F~ · dS~ = ZZ S1 F~ · ~n1dS = ZZ 06y6b 06z6c P (a, y, z)dydz. ZZ S2 F~ · dS~ = ZZ S2 F~ · ~n2dS = − ZZ 06y6b 06z6c P (0, y, z)dydz. òþ¡üªf\, 2/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ZZ S1+S2 F~ · dS~ = ZZ 06y6b 06z6c P (a, y, z) − P (0, y, z) dydz = ZZ 06y6b 06z6c Z a 0 ∂P ∂x dx dydz = ZZZ V ∂P ∂x dxdydz. Ónk ZZ S3+S4 F~ · dS~ = ZZZ V ∂Q ∂y dxdydz ZZ S5+S6 F~ · dS~ = ZZZ V ∂R ∂z dxdydz. u´ ZZ ∂V F~ · dS~ = ZZZ V ∂P ∂x + ∂Q ∂y + ∂R ∂z dxdydz. (11.1) 3/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ±e·òþ¡úªí2«þ. dÄn«a .«: A .« : V = {(x, y, z)| x1(y, z) 6 x 6 x2(y, z), (y, z) ∈ D} B .« : V = {(x, y, z)| y1(z, x) 6 y 6 y2(z, x), (z, x) ∈ D} C .« : V = {(x, y, z)| z1(x, y) 6 z 6 z2(x, y), (x, y) ∈ D} x y z A .« x y z B .« x y z D z = z1(x, y) z = z2(x, y) C .« 4/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© V ´ C .«, R(x, y, z) ∈ C1 (V ). S1 ´e.¡ z = z1(x, y), S2 ´þ.¡ z = z1(x, y). Ï V ý¡{ ~n z ¶R, ¤± ZZ ∂V Rdx ∧ dy = ZZ ∂V R~k · ~ndS = ZZ S2 Rdx ∧ dy + ZZ S1 Rdx ∧ dy = ZZ D R(x, y, z2(x, y))dxdy − ZZ D R(x, y, z1(x, y))dxdy = ZZ D R(x, y, z2(x, y)) − R(x, y, z1(x, y)) dxdy = ZZ D Z z2(x,y) z1(x,y) ∂R ∂z dz! dxdy = ZZZ V ∂R ∂z dxdydz. V ´dk C .«© ¤«, 3«ú¡þ, ü«>.{Ð, Ïd, 3q¡þÈ©-. u´k ZZ ∂V Rdx ∧ dy = ZZZ V ∂R ∂z dxdydz. 5/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© Ón, V ´dk A .«© ¤«, k ZZ ∂V P dy ∧ dz = ZZZ V ∂P ∂x dxdydz. V ´dk B .«© ¤«, k ZZ ∂V Qdz ∧ dx = ZZZ V ∂Q ∂y dxdydz. ù·Òy² V ©O©¤k A ., B ., C .«© , (11.1) E¤á. /, k ½n 1 (Gauss úª) m« V d©¡1wVýµ4¡ S ¤, S ý. e¼ê P, Q, R 3 V þkëY ê, Kk ZZ ∂V P dy∧dz+Qdz∧dx+Rdx∧dy = ZZZ V ∂P ∂x + ∂Q ∂y + ∂R ∂z dxdydz. (11.2) 6/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ~ 1 O I = ZZ S y(x − z)dydz + x 2dzdx + (y 2 + xz)dxdy, Ù¥ S ´N V = [0, a] × [0, b] × [0, c] Lý¡. ) Ï y(x − z), x2 , y2 + xz 3m¥´ C1 , ¤±d Gauss úª I = ZZZ V (y + x)dxdydz = ac Z b 0 ydy + bc Z a 0 xdx = acb2 2 + bca2 2 = abc(a + b) 2 . 7/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ~ 2 O A = ZZ S x dydz + y dzdx + z dxdy, Ù¥ S = {(x, y, z) : x, y, z > 0, x + y + z = 1}, { (1, 1, 1) Ó. ) V ´d S, S1, S2, S3 ¤¤o ¡N, ∂V {, ùp S1 = {(x, y, z) : x = 0, y > 0, z > 0, y + z 6 1} S2 = {(x, y, z) : x > 0, y = 0, z > 0, x + z 6 1} S3 = {(x, y, z) : x > 0, y > 0, z = 0, x + y 6 1} x y z S S S 1 2 3 d Gauss úª ZZ ∂V x dydz + y dzdx + z dxdy = ZZZ V 3dxdydz = 3 · 1 3 · 1 2 = 1 2 . ∴ ZZ S x dydz + y dzdx + z dxdy = 1 2 − ZZ S1 + ZZ S2 + ZZ S3 . ∵ ZZ S1 = 0, ZZ S2 = 0, ZZ S3 = 0, ∴ A = 1 2 . 8/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ~ 3 O A = ZZ S xy2 dydz + yz2 dzdx + zx2 dxdy, Ù¥ S ´¥N V = {(x, y, z) : x 2 + y 2 + z 2 6 z} L¡, . ) V ë꧴ x = r cos ϕ sin θ, y = r sin ϕ sin θ, z = 1 2 + r cos θ,(r, θ, ϕ) ∈ D, Ù¥ D : 0 6 r 6 1 2 , 0 6 θ 6 π, 0 6 ϕ 6 2π. d Gauss úª, A = ZZZ V (y 2 + z 2 + x 2 )dxdydz = ZZZ D (r 2 sin2 θ + 1 4 + r cos θ + r 2 cos2 θ)r 2 sin θ drdθdϕ = 2π ZZ06r6 1 2 06θ6π (r 2 + 1 4 + r cos θ)r 2 sin θdrdθ = 2π 1 5 · 1 2 5 · 2 + 1 4 · 1 3 · 1 2 3 · 2 = π 15 . 9/30 kJ Ik J I £ ¶ '4 òÑ
Gauss úª Stokes úª ©/ªÈ© ~ 4 O A = ZZ S (y 2 + z 2 ) dydz + (z 2 + x 2 ) dzdx + (x 2 + y 2 ) dxdy, Ù¥ S ´þ¥¡ {(x, y, z) : x 2 + y 2 + z 2 = a 2 , z > 0}, þ. ) - D ´ S 3 xy ²¡þÝK, =, x 2 + y 2 6 a 2 , D e, S D ¤¥N V. d Gauss úª ZZ S+D (y 2 + z 2 ) dydz + (z 2 + x 2 ) dzdx + (x 2 + y 2 ) dxdy = ZZZ V 0dxdydz = 0, ∴ A = ZZ −D (y 2 + z 2 ) dydz + (z 2 + x 2 ) dzdx + (x 2 + y 2 ) dxdy = ZZ x2+y26a2 (x 2 + y 2 )dxdy = ZZ 06r6a 06ϕ62π r 2 · rdrdϕ = 2π · 1 4 a 4 = πa4 2 . S D x y z 10/30 kJ Ik J I £ ¶ '4 òÑ