Question

consider the level surface generated by F(x,y,z)=0. Assume z=z(x,y), although we may never know this function explicitly. So we may assume that the surface in question is generated by F(x,y,z(x,y))=0. Using appropriate versions of the Chain Rule show:
1. ∂z/∂x=-(∂F/∂x)/(∂F/∂z)
2. ∂z/∂y=-(∂F/∂y)/(∂F/∂z)