已知函数f(x,y)具有二阶连续偏导数,且f(1,y)=0,f(x,1)=0,f(x,y)dxdy=a,其中D={(x,y)|0≤x≤1,0≤y≤1},计算二重积分I=xyf"xy(x,y)dxdy.

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问题 已知函数f(x,y)具有二阶连续偏导数,且f(1,y)=0,f(x,1)=0,f(x,y)dxdy=a,其中D={(x,y)|0≤x≤1,0≤y≤1},计算二重积分I=xyf"xy(x,y)dxdy.

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答案因为f(1,y)=0,f(x,1)=0,所以fy(1,y)=0,fx(x,1)=0. 从而 I=∫01xdx∫01yf(x,y)dy=∫01x[yfx(x,y)∫01一fx(x,y)dy]dx =一∫01dy∫01xfx(x,y)dx=一∫01[xf(x,y)|x=0x=1一∫01f(x,y)dx]dy =∫01dy∫01f(x,y)dx=a. a=∫01dy∫01f(x,y)dx =∫01[xf(x,y)|x=0x=1一∫01xfx(x,y)dx]dy =一∫01dx∫01xfx(x,y)dy =一∫01[xfx(x,y)=|y=0y=1一∫01xyfxy"(x,y)dy]dx =[*]xy fxy"(x,y)dσ 这里用到了条件f(1,y)=0,f(x,1)=0,并由此有fy(1,y)=0,fx(x,1)=0.

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