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本帖最后由 hbghlyj 于 2022-1-7 15:44 编辑 Ⅰ $x=x(y, z), y=y(x, z), z=z(x, y)$都是由方程$F(x, y, z)=0$所确定的具有连续偏导数的函数,证明\[\frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x}=-1\]
Ⅱ 设$y=f(x,t)$,而$t=t(x,y)$是由方程$F(x,y,t)=0$所确定的函数,其中$f,F$都具有一阶连续偏导数.试证明\[\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\frac{\partial f}{\partial x} \frac{\partial F}{\partial t}-\frac{\partial f}{\partial t} \frac{\partial F}{\partial x}}{\frac{\partial f}{\partial t} \frac{\partial F}{\partial y}+\frac{\partial F}{\partial t}}\] |
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