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)

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)

jamesj · Answer

Like the last problem I answered, as F is constant,
dF/dx = 0  and dF/dy = 0

Hence 
dF/dx = ∂F/∂x + ∂F/∂z . ∂z/∂x = 0 
which gives you the result in 1.  

Now repeat with dF/dy for 2.