text: 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