If $f(x,y,z)=0$ and $g(x,y,z)=0$, show that $$\frac{dx}{\frac{\partial (f,g)}{\partial (y,z)}}=\frac{dy}{\frac{\partial (f,g)}{\partial (z,x)}}=\frac{dz}{\frac{\partial (f,g)}{\partial (x,y)}}$$ under necessary conditions.
I tried wrote as $$df=f_xdx+f_ydy+f_zdz=0$$ and $$dg=g_xdx+g_ydy+g_zdz=0$$ also wrote out $$\frac{\partial(f,g)}{\partial(y,z)}=f_yg_z-f_zg_y$$ and so were others. But I still got stuck ... Any help? Thanks~
A direct approach: Let $v=(v_x,v_y,v_z)$ be in the common kernel: $df(v)=dg(v)=0$. Then $v$ is orthogonal to $\nabla f$ and $\nabla g$. So $\nabla f \times \nabla g= c v$. Write this down in coordiantes and you find the stated result (with $c$ being the common value in the equation when acting upon $v$).