CHAPTER VI 
APPLICATIONS TO THE GEOMETRY OF SPACE 
1. Tangent Plane and Normal Line to a Surface, (a) Explicit 
Form of the Equation of the Surface. Let the equation of the sur 
face be given in the explicit form, 
(!) 2 =/(*, y). 
Then the equation of the tangent plane at the point (ic 0 , y 0 , z 0 ) is 
(cf. Introduction to the Calculus, Chap. XV, § 3): 
(2) 
The equation of the normal line at the same point is 
(3) 
b> _ y — yp _ z — Zp 
dz 
hjo 
- 1 
Finally, the direction cosines of the normal at an arbitrary point 
(x, y, z) of (1) satisfy the relations : 
/*\ n dz dz * 
(4) cos a : cos (3 : cos y = — : — : — 1. 
dx dy 
(b) Implicit Form. If the equation of the surface is given in the 
implicit form, 
(5) F{x, y, z) = 0, 
it follows then from (2) and the expressions for the partial deriva 
tives, Chap. V, § 9, (8), that the equation of the tangent plane is 
(6) (f\ ( * “ 1 ^ + (f). 1[y ~ ya)+ (f) 0 (z “ 2o) " a 
For the normal line, 
and for the direction cosines of the normal at (x, y, z), 
(8) 
cos a : cos /3 : cos y = 
152 
8F ,d_F _dF 
dx dy dz