黎曼积分意义下重积分与累次积分不等同的例子

[]2003209226[](JYXM2003259)213Vol.21,.320056COLLEGEMATHEMATICSJun.2005,,,(,245021)[],,.,,.[];;;[]O172.2[]C[]167221454(2005)0320106204,:,.,,,..1,1f(x,y)=1qx+1qy,x,y,0,D=[0,1][0,1],qxqyxy,f(x,y)D,.f1(x,y)=1qx,x,y,0,,f2(x,y)=1qy,x,y,0,,f1(x,y),f2(x,y)D.,P(01),(0,1]1,x1,x2,,xN0xixi+11(i=1,2,,N-1).=4Nmin0iN-1(xi+1-xi)(x0=0,xN+1=1),Ik=[xk-,xk+](k=1,2,,N),Jk=[xk+,xk+1-](k=1,2,,N-1),J0=[0,x1-],JN=[xN+,1],DT:J0[0,1],I1[0,1],J1[0,1],I2[0,1],,JN[0,1],1,2,3,,2N+1.f1(x,y)i,Mi,mi,N,S(T)-s(T)=2N+1i=1(Mi-mi)i=Nk=0M2k+1Jk+Nk=1M2kIkNk=0Jk+Nk=11Ik+N22,f1(x,y)D.,f2(x,y)D.f(x,y)=f1(x,y)+f2(x,y)D.y,f(x,y)0;y,x,f(x,y)0;xf(x,y)=1qx+1qy,f(x,y)1qy0,f(x,y)x[0,1],xy.,yx.2,2f(x,y)=1,x,0y12;x,12y1,0,x,12y1;x,0y12D=[0,1][0,1],f(x,y)D,yx,xy.D=[0,1][0,1]T,f(x,y)Rkk=1(k=1,2,,n),f(x,y)D.,Px[0,1],x,10f(x,y)dy=120dy=12;x,10f(x,y)dy=112dy=12,10dx10f(x,y)dy=12,10dy10f(x,y)dx.,Py0[0,1],[0,1]T,i[xi-1,xi]i=1(i=1,2,,n),ni=1ixi=1,limnni=1ixi=10.f(x,y0)[0,1].xy.3,,3f(x,y)=x-y(x+y)3,(x,y)(0,0),0,(x,y)=(0,0)D=[0,1][0,1],f(x,y)D,,.7013,:f(x,y)(0,0).,2n,1n,f2n,1n=2n-1n2n+1n3=1n3n3=n227+(n),f(x,y)D=[0,1][0,1],10dx10x-y(x+y)3dy=10y(x+y)210dx=10dx(x+1)2=12,10dy10x-y(x+y)3dx=10-x(x+y)210dy=-10dy(y+1)2=-12,,.4,4f(x,y)=1,x=p1q,y=p2q,p1,p2,q,0,,f(x,y)D=[0,1][0,1].x0[0,1],x0,f(x0,y)=0;x0=p1q,fp1q,yqy=1q,2q,,qq,10dx10f(x,y)dy=0.10dy10f(x,y)dx=0.ni=1f(i,i)xiyi=1,(i,i)=p1q,p2q,0,(i,i),f(x,y)D,kDf(x,y)dxdy.,,?,,;,kDf(x,y)dxdy(D=[a,b][c,d]),x[a,b],I(x)=dcf(x,y)dy,badxdcf(x,y)dy,kDf(x,y)dxdy=badxdcf(x,y)dy.,.801215f(x,y)=1p,x,y,x=qp,0,D=[0,1][0,1],f(x,y)D,10dy10f(x,y)dx,kDf(x,y)dxdy=10dy10f(x,y)dx10dx10f(x,y)dy.(i)f(x,y)D,1;(ii)x(0,1)p,f(x,y)[0,1],.,p=nm,m,n,f(p,y)=1m,y,0,y.[0,1],f(p,y)1m,f(x,y)[0,1],.,,.,(),,,().,,..,,.[][1].[M].:,1991.[2],.[M].:,1986.AnalysisontheMultipleIntegralandRepeatedIntegralwithInstancesXIANGMing2yin,YEMing,FANGJi2guang,BAOZhi2hui(DepartmentofMathematicsHuangshanCollege,Huangshan,Anhui245021,China)Abstract:Thetheoryofmultipleintegralisbasicallyparalleltothatofdefinitionandmainnature.Butbecauseofthechargesofthespaceframe,theyalsopresenttheessentialdistinction.Thispaperusingseveralinstances,analysesthatthemultipleintegralandrepeatedintegralaretwoindependentconceptsandthereisnocertainimplicationbetweenthem.Italsopointsoutthatonlywhentheymeetsomecertainrequirementscantheybeequaltoeachother.Keywords:multipleintegral;repeatedintegral;existence;relation9013,: