u$ = \frac{yz}{x} \; \therefore \frac{\partial u}{\partial x} \; = \frac{-yz}{x^2} \; , \; \frac{\partial u}{\partial y} \; = \frac{z}{x} \; , \; \frac{\partial u}{\partial z} \; = \frac{y}{x} \\ \; \\ \; \\ v = \frac{zx}{y} \; \therefore \frac{\partial v}{\partial x} \; = \frac{z}{y} \; , \; \frac{\partial v}{\partial y} \; = \frac{-zx}{y^2} \; , \; \frac{\partial v}{\partial z} \; = \frac{x}{y} \\ \; \\ \; \\ w = \frac{xy}{z} \; \therefore \frac{\partial w}{\partial x} \; = \frac{y}{z} \; , \; \frac{\partial w}{\partial y} \; = \frac{x}{z} \; , \; \frac{\partial w}{\partial z} \; = \frac{-xy}{z^2} \\ \; \\ \; \\ J \Big( \frac{u,v,w}{x,y,z} \Big) \;= \; \left| \begin{array}{ccc} \dfrac{\partial u }{\partial x} & \dfrac{\partial u }{\partial y} & \dfrac{\partial u }{\partial z} \\ \; \\ \dfrac{\partial v }{\partial x} & \dfrac{\partial v }{\partial y} & \dfrac{\partial v }{\partial z} \\ \; \\ \dfrac{\partial w }{\partial x} & \dfrac{\partial w }{\partial y} & \dfrac{\partial w }{\partial z} \end{array} \right| \;= \; \left| \begin{array}{ccc} \frac{-yz}{x^2} & \frac{z}{x} & \frac{y}{x} \\ \; \\ \frac{z}{y} & \frac{-zx}{y^2} & \frac{x}{y} \\ \; \\ \frac{y}{z} & \frac{x}{z} & \frac{-xy}{z^2} \end{array} \right| \\ \; \\ \; \\ = \frac{-yz}{x^2} \Big[ (\frac{-zx}{y^2}) (\frac{-xy}{z^2}) \; - \; \frac{x^2}{yz} \big] \; - \; \frac{z}{x} \Big[ \frac{z}{y}(\frac{-xy}{z^2}) \; - \; \frac{x}{y} \cdot \frac{y}{z} \Big] \; + \; \frac{y}{z} \Big[ \frac{z}{y} \cdot \frac{x}{z} \; - \; (\frac{-zx}{y^2}) \frac{y}{z} \Big] \\ \; \\ = \frac{-yz}{x^2} \Big[ \frac{x^2}{yz} \; - \; \frac{x^2}{yz} \big] \; - \; \frac{z}{x} \Big[ \frac{-x}{z} \; - \; \frac{x}{z} \Big] \; + \; \frac{y}{z} \Big[ \frac{x}{y} \; + \; \frac{x}{y} \Big] \\ \; \\ \; \\ J \Big( \frac{u,v,w}{x,y,z} \Big) \; = \; 2 \; + \; 2 \; = \; 4 $