2. 考研
  3. 设f(x，y)在[a，b]×[c，d]上连续，g(x，y)=∫0xdu∫cyf(u，v)dv，证明：g’’xy=g’’yx=f(x，y)(a＜x＜b，c＜y＜d)．

设f(x，y)在[a，b]×[c，d]上连续，g(x，y)=∫0xdu∫cyf(u，v)dv，证明：g’’xy=g’’yx=f(x，y)(a＜x＜b，c＜y＜d)．

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问题 设f(x，y)在[a，b]×[c，d]上连续，g(x，y)=∫0xdu∫cyf(u，v)dv，证明：g’’xy=g’’yx=f(x，y)(a＜x＜b，c＜y＜d)．
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答案由于 [*] 又g(x，y)=∫axdu∫cxf(u，v)=∫cydv∫ax(u，v)du，故g”xy=f(x,y)，所以g”xy=g”yx=f(x,y)．
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考研数学二

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