Originally Posted by
vincisonfire $\displaystyle
\frac{dG}{dx} = \frac{\partial G}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial G}{\partial y}\frac{\partial y}{\partial x} +\frac{\partial G}{\partial z}\frac{\partial z}{\partial x}
$
$\displaystyle
0 = \frac{\partial G}{\partial x}\cdot 0+\frac{\partial G}{\partial y}\frac{\partial y}{\partial x} +\frac{\partial G}{\partial z}\frac{\partial Z}{\partial x}
$
$\displaystyle
\frac{- \frac{\partial G}{\partial x}}{\frac{\partial G}{\partial y}}=\frac{\partial y}{\partial x}
$
Similarly
$\displaystyle
\frac{- \frac{\partial G}{\partial y}}{\frac{\partial G}{\partial z}}=\frac{\partial z}{\partial y}
$
$\displaystyle
\frac{- \frac{\partial G}{\partial z}}{\frac{\partial G}{\partial x}}=\frac{\partial x}{\partial z}
$
$\displaystyle
\frac{- \frac{\partial G}{\partial x}}{\frac{\partial G}{\partial y}}\cdot \frac{- \frac{\partial G}{\partial y}}{\frac{\partial G}{\partial z}} \cdot \frac{- \frac{\partial G}{\partial z}}{\frac{\partial G}{\partial x}}=-1
$
See that each term at the numerator eliminates with one at the denominator.