⑩天掌 Teaching Plan on Advanced Mathematics o 证明(1) y=2(x) 若区域D既是X一型 又是y-型,即平行于 x=yl(y D B 坐标轴的直线和L至 多交于两点 x=v2(y) y=p(r) X D={(x,y)q(x)≤p≤q(x),a≤x≤b} D={(x,y)v(y)≤x≤v2()e≤ys tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics {( , ) ( ) ( ), } D = x y 1 x y 2 x a x b 证明(1) 若区域D既是X −型 又是Y −型,即平行于 坐标轴的直线和L至 多交于两点. {( , ) ( ) ( ), } D = x y 1 y x 2 y c y d y x o a b D c d ( ) y = 1 x ( ) y = 2 x A B C E ( ) 2 x = y ( ) 1 x = y
⑩天掌 Teaching Plan on Advanced Mathematics o 00 v)00 drdy= dyj d x D a rOO), y)dy-o(,(v),y)dy =n(x,y)- Q(x,y)小y x=yuly CAE mQ(x,y)+」Q(xy x=v2(y) X Q(x,y)小y 同理可证 aP dxdy=t P(x, y)dx tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics dx x Q dxdy dy x Q y y d c D = ( ) ( ) 2 1 = − d c d c Q( ( y), y)dy Q( ( y), y)dy 2 1 = − CBE CAE Q(x, y)dy Q(x, y)dy = + CBE EAC Q(x, y)dy Q(x, y)dy = LQ(x, y)dy 同理可证 = − L D dxdy P x y dx y P ( , ) y x o d ( ) 2 x = y D c C E ( ) 1 x = y
⑩天掌 Teaching Plan on Advanced Mathematics o 00 oP 两式相加得 h=Px+Q小 证明(2) 若区域D由按段光 滑的闭曲线围成如图, 将D分成三个既是X一型又是 Y-型的区域D1,D2,D3 00 aP 00 aP )dxdy ax a D,+, +D, ax g dta tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics 若区域D由按段光 滑的闭曲线围成.如图, 证明(2) L L1 L2 L3 D D1 D2 D3 两式相加得 = + − L D dxdy Pdx Qdy y P x Q ( ) 将D分成三个既是X −型又是 Y −型的区域D1 ,D2 ,D3 . + + − = − 1 2 3 ( ) ( ) D D D D dxdy y P x Q dxdy y P x Q
⑩天掌 Teaching Plan on Advanced Mathematics o a0 aP d0 aP )dxdy+ )dxdy ax a )dxdy D D2 Pax+2dy+ pd+gdy+h pdx+ody Px+Q小 公 (L1L2,L23对D来说为正方向) tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics − + − + − 1 2 3 ( ) ( ) ( ) D D D dxdy y P x Q dxdy y P x Q dxdy y P x Q = + + + + + L1 L2 L3 Pdx Qdy Pdx Qdy Pdx Qdy = + L Pdx Qdy D1 D2 D3 L L1 L2 L3 ( , ) L1, L2 L3对D来说为正方向
⑩天掌 Teaching Plan on Advanced Mathematics o 证明(3) 若区域不止由一条闭曲 线所围成添加直线段ABCE 则D的边界曲线由ABL2,BA AFC. CE. L3,BC及CGA构成 C 由(2)知∫ a0 aP D )dardy F ax ay A ={++|+++ ABdL d BAJAFCUCE ECJCGA }·(Px+g小y) +5,+5,)(P+Q小y) tianjin polytechnic dmivendity
Tianjin Polytechnic University Teaching Plan on Advanced Mathematics G D L3 L2 F C E L1 A B 证明(3) 若区域不止由一条闭曲 线所围成.添加直线段 AB,CE. 则D的边界曲线由AB,L2,BA, AFC,CE, L3, EC 及 CGA 构成. 由(2)知 − D dxdy y P x Q ( ) = + + + + AB L2 BA AFC CE { + + + + L EC CGA } (Pdx Qdy) 3 = + + + 2 3 1 ( )( ) L L L Pdx Qdy