第二类曲面积分的计算方法

X(464000)O172.2(),1(()2023)k(z2+x)dydz-zdxdy,z=12(x2+y2)z=0z=2kPdydz+Qdzdx+Rdxdy=k(Pcos+Qcos+Rcos)dS(1)cos,cos,cos(x,y,z)ˆn={x,y,-1},ˆn0={cos,cos,cos}=x1+x2+y2,y1+x2+y2,-11+x2+y2k(z2+x)dydz-zdxdy=k(z2+x)x1+x2+y2-z-11+x2+y2dS=kz2x+x2+z1+x2+y2dS=kx2+z1+x2+y2dS=kDx2+12(x2+y2)1+x2+y21+x2+y2dxdy=kDx2+12(x2+y2)dxdy=20d20r2cos2+r22rdr=8z=z(x,y),kR(x,y,z)dxdy=kDxyR[x,y,z(x,y)]dxdy(2)x=y(y,z),kP(x,y,z)dydz=kDyzP[x(y,z),y,z)]dydz(3)y=y(z,x),kQ(x,y,z)dzdx=kDzxQ[x,y(z,x),z]dzdx(4):z=z(x,y)(x=x(y,z),y=y(z,x))23STUDIESINCOLLEGEMATHEMATICSVol.7,No.2Mar.,2004X:2002-09-13©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(),;,z=z(x,y)(x=x(y,z),y=y(z,x))(,),k(z2+x)dydz-zdxdy=k(z2+x)dydz-kzdxdyk(z2+x)dydz=k(z2+x)dydz+k(z2+x)dydz=kDyz(z2+2z-y2)dydz-kDyz(z2-2z-y2)dydz=2kDyz2z-y2dydz=420dy2y222z-y2dz=4kzdxdy=-12kDxy(x2+y2)dxdy=-1220d20r3dr=-4k(z2+x)dydz-zdxdy=8,,,,,z=z(x,y),(x,y)Dxy(DxyxOy),P,Q,P,ƒen={cos,cos,cos}=-zxz2x+z2y+1,-zyz2x+z2y+1,1z2x+z2y+1dydz=cosdS,dzdx=cosdS,dxdy=cosdS,dydz=cosdS=coscoscosdS=coscosdxdy=-zxdxdydzdx=cosdS=coscoscosdS=coscosdxdy=-zydxdykP(x,y,z)dydz+Q(x,y,z)dzdx+R(x,y,z)dxdy=kDxy{P[x,y,z(x,y)][-zx(x,y)]+Q[x,y,z(x,y)][-zy(x,y)]+R[x,y,z(x,y)]}dxdy=kDxy[P(-zx)+Q(-zy)+R]dxdy(5):z=z(x,y),,y=y(z,x)x=x(y,z),{-yx,1,-yz}{1,-xy,-xz},(5),(37)3372:©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.[1].k-1r=1rn.,1992,2(2):17-18.(33)z=12(x2+y2),xOy:Dxy={(x,y)|x2+y2F4},,zx=x,(5)k(z2+x)dydz-zdxdy=-kDxy14(x2+y2)2+x(-x)-12(x2+y2)dxdy=-kDxyx2+12(x2+y2)dxdy=20d20[r2cos2+r22]rdr=8lPdydz+Qdzdx+Rdxdy=m5P5x+5Q5y+5R5zd(6),,1z=2,k+1(z2+x)dydz-zdxdy=m0d=0k(z2+x)dydz-zdxdy=-k1(z2+x)dydz-zdxdyk1(z2+x)dydz-zdxdy=0-k1zdxdy-2kDxydxdy=-8k(z2+x)dydz-zdxdy=8(34)341-1e[10e-x2dx]2I=10e-x2dx1I2=10e-x2dx10e-y2dy=kD1e-x2-y2dxdykD2e-x2-y2dxdy,D1:0FxF1,0FyF1;D2:x2+y2F1,xE0,yE0,kD2e-x2-y2dxdy=20d10e-r2rdr=41-1e,I2=10e-x2dx241-1e17372:©1994-2006ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.