From calculus it is known that the gradient vector  is normal to the curve  (see e.g. Level set#Level sets versus the gradient). If it is shown that everywhere  using the formula for  in terms of  then this proves that level curves of  are streamlines. 
;Cylindrical coordinates:
In cylindrical coordinates,
and  
So that
;Spherical coordinates:
And in spherical coordinates
and 
So that