性质Green公式曲线定向第二型曲线积分设π=（a,y，z），则dr=（dac,dy，dz）.于是第二型曲线积分也常表示为Pda +Qdy + Rdz.(11.2)L定理1设有区域D CR3,连续向量场F：D→R3.又设LCD是一条有向光滑曲线,其参数方程为π=(t),α≤t≤β,并且参数增加的方向为L的方向，则有F. dr=(Fo) .T(t)dt.(11.3)JLa若 F = (P, Q, R), r(t) = (r(t), y(t), z(t), 则F . dr = / (P(e(t), g(t),z(t)a(t)L+ Q(c(t), y(t), z(t)y'(t) + R(a(t), y(t), z(t)z(t))dt. (11.4)返回全屏关闭退出6/32
½ 1.È© 5 Green úª ~r = (x, y, z), K d~r = (dx, dy, dz). u´1.È©~L« Z L P dx + Qdy + Rdz. (11.2) ½n 1 k« D ⊂ R3 , ëYþ| F~ : D → R3 . q L ⊂ D ´^ k1w, Ùëê§ ~r = ~r(t), α 6 t 6 β, ¿ ëêO\ L , Kk Z L F~ · d~r = Z β α (F~ ◦ ~r) · ~r0 (t) dt. (11.3) e F~ = (P, Q, R), ~r(t) = (x(t), y(t), z(t)), K Z L F~ · d~r = Z β α P (x(t), y(t), z(t))x 0 (t) + Q(x(t), y(t), z(t))y 0 (t) + R(x(t), y(t), z(t))z 0 (t) dt. (11.4) 6/32 kJ Ik J I £ ¶ '4 òÑ
性质曲线定向Green公式第二型曲线积分证明 令(t) = (a(t),y(t),z(t)), F= (P,Q,R). 设定义中的点列 [)为(r(ti)),α=to<ti<.<tn=β.弧段ri-iri上的点si=(i),ti-1≤Ti≤ti.因此nF(s) . Ar; =For(T) . (r(t:) - r(ti-1)i=1i=1nZ P(r(T:)(a(t:) - (t-1) + Q(T:)(t) - y(t=-1)1iW1+R(r(Ti))(z(ti) - z(ti-1))1由微分中值定理，存在入i，MiViE[ti-1,t]使得c(ti)-α(ti-1)= α'(^;)Ati,y(ti) -y(ti-1) = y'(μi)Ati,z(ti) -z(ti-1) = z'(vi)△ti,返回全屏关闭退出7/32
½ 1.È© 5 Green úª y² - ~r(t) = (x(t), y(t), z(t)), F~ = (P, Q, R). ½Â¥: {~r} {~r(ti)}, α = t0 < t1 < · · · < tn = β. lã ) ri−1ri þ: ξi = ~r(τi), ti−1 6 τi 6 ti. Ïd X n i=1 F~(ξi) · ∆~ri = X n i=1 F~ ◦ ~r(τi) · (~r(ti) − ~r(ti−1)) = X n i=1 P (~r(τi))(x(ti) − x(ti−1)) + X n i=1 Q(~r(τi))(y(ti) − y(ti−1)) + X n i=1 R(~r(τi))(z(ti) − z(ti−1)). d©¥½n, 3 λi, µi, νi ∈ [ti−1, ti ] ¦ x(ti) − x(ti−1) = x 0 (λi)∆ti, y(ti) − y(ti−1) = y 0 (µi)∆ti, z(ti) − z(ti−1) = z 0 (νi)∆ti, 7/32 kJ Ik J I £ ¶ '4 òÑ
性质曲线定向第二型曲线积分Green公式因此F(si) . Ari =P(r(Ti)a'(A;)△ti +Z Q(r(Ti)y'(ui)Ati1i=1i-1i-1nZ+R(r(Ti))z'(vi)△tii=1令maxtil→0,就得到F.dr=(P(c(t), y(t), z(t))ac'(t)T+ Q(r(t), y(t), z(t)y'(t) + R(r(t), y(t), z(t)z'(t))dt.注意，对于形如「Pdac的积分，应该理解成是一个特殊的向量场F=（P,0,0)的第二型曲线积分,而不是通常的定积分，这里的dac 是有向弧长元在轴的投影.其他情形类似返回全屏关闭退出18/32
½ 1.È© 5 Green úª Ïd X n i=1 F~(ξi) · ∆~ri = X n i=1 P (~r(τi))x 0 (λi)∆ti + X n i=1 Q(~r(τi))y 0 (µi)∆ti + X n i=1 R(~r(τi))z 0 (νi)∆ti. - max i |∆ti| → 0, Ò Z L F~ · d~r = Z β α P (x(t), y(t), z(t))x 0 (t) + Q(x(t), y(t), z(t))y 0 (t) + R(x(t), y(t), z(t))z 0 (t) dt. 5¿, éu/X R L P dx È©, ATn)¤´AÏþ| F~ = (P, 0, 0) 1.È©, Ø´Ï~½È©, ùp dx ´kl 3 x ¶ÝK. Ù¦/aq. 8/32 kJ Ik J I £ ¶ '4 òÑ
性质曲线定向第二型曲线积分Green公式第二型曲线积分具有以下性质1）线性：即若F=CiF+C2F2,则有F. dr = c1 F·dr+ C2F2 . dr.特别,当 =Pi, =Q, = R时,=++ =Pi+Q+Rk因此F.dr:Pda +Qdy + Rdz1Pdc +RdzQdy +JLLL2）对积分曲线的可加性：若LAc是由LAB和LBc连接而成的，则有F.dr.F.dr=F.dr+JLABJLBCLAC所以对于逐段光滑的曲线，可以分段进行积分3）积分的方向性：JLaF.dr=一JupF.drI-I返回全屏关闭退出9/32
½ 1.È© 5 Green úª 1.È©äk±e5: 1) 5: =e F~ = c1F~1 + c2F~2, Kk Z L F~ · d~r = c1 Z L F~1 · d~r + c2 Z L F~2 · d~r. AO, F~1 = P~i, F~2 = Q~j, F~3 = R~k , F~ = F~1 + F~2 + F~3 = P~i + Q~j + R~k. Ïd Z L F~ · d~r = Z L P dx + Qdy + Rdz = Z L P dx + Z L Qdy + Z L Rdz 2) éÈ©\5: e LAC ´d LAB Ú LBC ë ¤, Kk Z LAC F~ · d~r = Z LAB F~ · d~r + Z LBC F~ · d~r. ¤±éuÅã1w, ±©ã?1È©. 3) È©5: R LAB F~ · d~r = − R LBA F~ · d~r. 9/32 kJ Ik J I £ ¶ '4 òÑ
性质曲线定向Green公式第二型曲线积分例1计算曲线积分fiayda+ady，L是三角形OAB的正向周界，其中 A = (1, 2), B = (0, 2).解cyda + a'dy=cyda + a'dy.T14r'dcyd + a'dy =(α·2+22)d=43JLIJOcyda +α?dy=-2(1 - α)d = -1,JL2cyd + αdy = 0JL3BA所以Vcyda+r'dy33返回退出全屏关闭10/32
½ 1.È© 5 Green úª ~ 1 OÈ© R L xydx + x 2dy, L ´n/ OAB ±., Ù ¥ A = (1, 2), B = (0, 2). ) Z L xydx + x 2dy = Z L1 + Z L2 + Z L3 xydx + x 2dy. Z L1 xydx + x 2dy = Z 1 0 (x · 2x + 2x 2 )dx = 4 Z 1 0 x 2dx = 4 3 Z L2 xydx + x 2dy = −2 Z 1 0 (1 − x)dx = −1, Z L3 xydx + x 2dy = 0 ¤± Z L xydx + x 2dy = 4 3 − 1 = 1 3 . O x y B A L L L 1 2 3 10/32 kJ Ik J I £ ¶ '4 òÑ