Question Number 176374 by mathocean1 last updated on 17/Sep/22
$${Show}\:: \\ $$$$\frac{\partial{X}}{\partial{Y}}\mid_{{Z}} \frac{\partial{Y}}{\partial{Z}}\mid_{{X}} \frac{\partial{X}}{\partial{Z}}\mid_{{Y}} =−\mathrm{1} \\ $$
Answered by aleks041103 last updated on 17/Sep/22
$${dX}=\frac{\partial{X}}{\partial{Y}}\:{dY}\:+\:\frac{\partial{X}}{\partial{Z}}\:{dZ} \\ $$$${dY}=\frac{\partial{Y}}{\partial{Z}}\:{dZ}+\frac{\partial{Y}}{\partial{X}}\:{dX} \\ $$$${dZ}=\frac{\partial{Z}}{\partial{X}}\:{dX}\:+\:\frac{\partial{Z}}{\partial{Y}}\:{dY} \\ $$$$\Rightarrow{dX}=\frac{\partial{X}}{\partial{Y}}\left(\frac{\partial{Y}}{\partial{Z}}\:{dZ}+\frac{\partial{Y}}{\partial{X}}\:{dX}\right)+\:\frac{\partial{X}}{\partial{Z}}\:{dZ}= \\ $$$$=\left(\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{Z}}\:+\frac{\partial{X}}{\partial{Z}}\right){dZ}+\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{X}}\:{dX} \\ $$$${But}\:\frac{\partial{X}}{\partial{Y}}=\frac{\mathrm{1}}{\frac{\partial{Y}}{\partial{X}}}\Rightarrow\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{X}}=\mathrm{1} \\ $$$$\Rightarrow\left(\mathrm{1}−\mathrm{1}\right){dX}=\left(\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{Z}}\:+\frac{\partial{X}}{\partial{Z}}\right){dZ}=\mathrm{0} \\ $$$$\Rightarrow\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{Z}}\:+\frac{\partial{X}}{\partial{Z}}=\mathrm{0} \\ $$$$\Rightarrow\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{Z}}\:=−\frac{\partial{X}}{\partial{Z}} \\ $$$$\Rightarrow\frac{\partial{X}}{\partial{Y}}\:\frac{\partial{Y}}{\partial{Z}}\:\frac{\partial{Z}}{\partial{X}}=−\mathrm{1} \\ $$
Commented by mathocean1 last updated on 01/Oct/22
$${thanks} \\ $$