来自陈曙玲的问题
设f(x,y)在xOy面连续,且F(t)=∫∫f(x,y)dxdy,其中D为x^2+y^2=t^2所围的区域,求lim(t+趋于0)F‘(t)/t
设f(x,y)在xOy面连续,且F(t)=∫∫f(x,y)dxdy,其中D为x^2+y^2=t^2所围的区域,求lim(t+趋于0)F‘(t)/t
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2020-05-13 00:34
设f(x,y)在xOy面连续,且F(t)=∫∫f(x,y)dxdy,其中D为x^2+y^2=t^2所围的区域,求lim(t+趋于0)F‘(t)/t
设f(x,y)在xOy面连续,且F(t)=∫∫f(x,y)dxdy,其中D为x^2+y^2=t^2所围的区域,求lim(t+趋于0)F‘(t)/t
∵F(t)=∫∫f(x,y)dxdy=∫dθ∫f(rcosθ,rsinθ)rdr(应用极坐标变换)∴F'(t)=[∫dθ∫f(rcosθ,rsinθ)rdr]'=∫[∫f(rcosθ,rsinθ)rdr]'dθ(∵f(x,y)在xOy面连续)=∫f(tcosθ,tsinθ)tdθ=t∫f(tcosθ,tsinθ)dθ...