Vg \gÈ© §10.3 nÈ© 10.3.1 nÈ©Vg V = [a1, b1] × [a2, b2] × [a3, b3] ´ R3 ¥n4«m. ©O Ii = [ai, bi ] (i = 1, 2, 3) þ©: T1 : a1 = x0 < x1 < · · · < xn = b1; T2 : a2 = y0 < y1 < · · · < ym = b2; T3 : a3 = z0 < z1 < · · · < zl = b3. nx²1²¡ x = xi, (i = 0, 1, · · · , n), y = yj, (j = 0, 1, · · · , m) z = zk (k = 0, 1, · · · , l) r V ©¤ n × m × l f«m: Vijk = [xi−1, xi ] × [yj−1, yj] × [zk−1, zk] (i = 1, 2, · · · , n; j = 1, 2, · · · , m; k = 1, 2, · · · , l.) ù f«m|¤ V © T = T1 × T1 × T3. éu3 V þ½Â¼ê 1/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© f(x, y, z), 3z Vijk ¥: ξijk, Úª (Riemann Ú) S(f, T ) := X n i=1 X m j=1 X l k=1 f(ξijk)σ(Vijk), (10.1) Ù¥ σ(Vijk) ´ Vijk NÈ. P kT k = max i,j,k {diam(Vijk)}, ùp diam(Vijk) ´ Vijk éÝ, ¡ kT k © T °Ý. ¡ ξijk :. ½Â 1 f(x, y, z) ´½Â3 V þ¼ê. XJ3ê A, ¦é?¿ ½ ε > 0, 3 δ > 0, kT k < δ , ØØ: ξijk 3 Vijk ¥XÛÀ, Ñk X n i=1 X m j=1 X l k=1 f(ξijk)σ(Vijk) − A < ε, K¡¼ê f 3«m V þÈ, ¿ò A ZZZ V f(x, y, z)dxdydz ½ Z V fdσ, ¡ f 3«m V þnÈ©. 2/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© ½Â 2 V ⊂ R3 ´8Ü. XJé?¿½ ε > 0, 3k ½n«m {In, n ∈ N} ¦ V ⊂ [ n∈N In, X ∞ n=1 σ(In) 6 ε, (ùp σ(In) L« In NÈ), @o¡ V (n) "ÿÝ8, {¡"ÿ8. e þ¡ In Ik, K V ¡"NÈ8. ½n 1 (1) õê8´"ÿ8, õê"ÿ8¿8´"ÿ8; (2) k"NÈ8¿8´"NÈ8; (3) B ´"NÈ8 du B ´"NÈ8; (4) B ⊂ R3 ´k.48. K B ´"ÿ8 du B ´"NÈ8; (5) R3 ¥1w¡¡´"NÈ8. ½n 2 (Lebesgue ½n) f(x, y, z) ´n4«m I ⊂ R3 þk.¼ ê, @o f 3 I þÈ¿©7^´: f mä: N´"ÿ8. 3/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© f(x, y, z) ´½Â3k.8 V ⊂ R3 þ¼ê. Ue¡ªò§ò ÿ R3 ¼ê: - fV (x, y, z) = f(x, y, z), (x, y, z) ∈ V ; 0, (x, y, z) 6∈ V, ½Â 3 f(x, y, z) ´½Â3k.8 V ⊂ R3 þ¼ê. I ´n «m, V ⊂ I. e fV (x, y, z) 3 I þÈ, K¡ f 3 V þÈ. È©Ò ´ fV (x, y, z) 3 I þÈ©, P ZZZ V f(x, y, z) dxdydz ½ Z V fdσ. nÈ©AÛ¿ÂØ*. ±ÄÔn¿Â: m¥ÔNÓ k« V ⊂ R3 , ÔNÝ¼ê´ ρ(x, y, z), Ï NÈ dσ þ´ ρ(x, y, z)dσ. u´ÔnoþÒ´È© Z V ρdσ. 4/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© ½Â 4 (k.8NÈ) V ⊂ R3 ´k.8. e 1 ~¼ê 3 V þÈ, K¡ V ´kNÈ8, ÙNȽ σ(V ) = R V 1 dσ. ½n 3 V ⊂ R3 ´k.8. K V ´"NÈ8 = V kNÈ NÈ ", = σ(V ) = Z V 1 dσ = 0. ½n 4 V ⊂ R3 ´k.8. K V kNÈ = ∂V ´"NÈ8. ½n 5 V ⊂ R3 ´kNÈ8. K f 3 V þÈ È© u A ¿© 7^´: é ∀ ε > 0, ∃ δ > 0, eò V ©kpØUkNÈ ¬ V1, · · · , Vn, P λi Vi ». ©°Ý λ = max 16i6n λi ÷v λ < δ, @oé ∀ pi ∈ Vi , Ñk X n i=1 f(pi)σ(Vi) − A < ε. 5/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© 10.3.2 nÈ©\gÈ© ½n 6 f(x, y, z) 3 V = I1 × I2 × I3 (Ii = [ai, bi ], i = 1, 2, 3) þ È. 1 ◦ XJéz (y, z) ∈ I2 × I3, f(x, y, z) ´ I1 þ'u x ȼê, K È© ϕ(y, z) = R b1 a1 f(x, y, z)dx ½Â 'uCþ (y, z) 3 I2 × I3 þÈ ¼ê, ¿k ZZ I2×I3 ϕ(y, z)dydz = ZZ I2×I3 dydz Z b1 a1 f(x, y, z)dx = ZZZ V f(x, y, z)dxdydz. 2 ◦ Ón, XJéz x ∈ [a1, b1], f(x, y, z) ´ I2 × I3 þ'u (y, z) ȼê, K ψ(x) = RR I2×I3 f(x, y, z)dy ´'u x 3 [a1, b1] þȼê, ¿k Z b1 a1 ψ(x)dx = Z b1 a1 dx ZZ I2×I3 f(x, y, z)dydz = ZZZ V f(x, y, z)dxdydz. 6/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© ½n 7 (\gÈ©, k) V ⊂ R3 ´kNÈk.8, f ´ V þë Y¼ê. V 3 xy ²¡þRÝK D, (x, y) ∈ D , Lù : Ru xy ²¡ V ¤«m [ϕ1(x, y), ϕ2(x, y)], @ok Z V fdσ = ZZ D dxdy Z ϕ2(x,y) ϕ1(x,y) f(x, y, z)dz. (10.2) O x y z D V (x, y) ϕ1(x, y) ϕ2(x, y) 7/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© y² Ï f 3 V þëY, ¤± f 3 V þÈ. n«m I = I1 × I2 × I3 ⊃ V, Ù¥ I1, I2 Ú I3 Ñ´ R ¥4«m. K fV 3 I þÈ, Z V fdσ = Z I fV dσ. (x, y) ∈ D , ¼ê f(x, y, ·) 3«m [ϕ1(x, y), ϕ2(x, y)] þëY, l ´ È. Ïd Z I3 fV (x, y, z)dz = Z ϕ2(x,y) ϕ1(x,y) f(x, y, z)dz, du (x, y) 6∈ D , fV (x, y, z) = 0, ù R I3 f(x, y, z)dz = 0. Ïd Z V fdσ = Z I fV dσ = ZZ I1×I2 dxdy Z I3 fV (x, y, z)dz = ZZ D dxdy Z ϕ2(x,y) ϕ1(x,y) f(x, y, z)dz. 8/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© ½n 8 (\gÈ©, k) V ⊂ R3 ´kNÈk.8, f ´ V þë Y¼ê. V 3 z ¶þÝK«m [c, d] z ∈ [c, d] , Lù: Ru z ¶²¡ V ¤ã/3 xy ²¡þÝK Dz, @ok Z V fdσ = Z d c dz ZZ Dz f(x, y, z)dxdy. (10.3) O x y z c d z V Dz 9/25 kJ Ik J I £ ¶ '4 òÑ
Vg \gÈ© ~ 1 ¦ ZZZ D x 2yexyzdxdydz, Ù¥ D = [0, 1]3 . ) ZZZ D x 2yexyzdxdydz = ZZ [0,1]2 x 2ydxdy Z 1 0 e xyzdz = ZZ [0,1]2 x(e xy − 1)dxdy = Z 1 0 xdx Z 1 0 e xydy − Z 1 0 xdx Z 1 0 dy = Z 1 0 (e x − 1)dx − 1 2 = e − 5 2 . 10/25 kJ Ik J I £ ¶ '4 òÑ