If the surface is expressed in an implicit form F(x, y, z ) = 0 , then a unit normal
to the surface can be obtained from the relation:
ˆen= grad F
|grad F|
.
If the surface is expressed in the explicit form z=z(x, y ),then a unit normal to
the surface can be found from the relation
ˆen= grad [z(x, y )−z]
|grad [z(x, y )−z]|
=
∂z
∂x ˆe^1 +
∂z
√ ∂y eˆ^2 −eˆ^3
1 +
(∂z
∂x
) 2
+
(
∂z
∂y
) 2 (7 .86)
Surfaces can also be expressed in the parametric form
x=x(u, v ), y =y(u, v ), z =z(u, v ),
where uand vare parameters. The functions x(u, v ), y (u, v ),and z(u, v )must be such
that one and only one point (u, v ) maps to any given point on the surface. These
functions are also assumed to be continuous and differentiable. In this case, the
position vector to a point on the surface can be represented as
r =r (u, v ) = x(u, v )ˆe 1 +y(u, v )ˆe 2 +z(u, v )ˆe 3.
The curves
r (u, v )
v=Constant
and r (u, v )
u=Constant
sweep out coordinate curves on the surface and the vectors
∂r
∂u ,
∂r
∂v
are tangent vectors to these coordinate curves. A unit normal to the surface at a
point P on the surface can then be calculated from the cross product of the tangent
vectors tangent vectors ∂r∂u and ∂r∂v evaluated at the point P. One can calculate the
unit normal
ˆen=
∂r
∂v ×
∂r
∣ ∂u
∣∂u∂r ×∂r∂v ∣∣.
It should be noted that if the cross product
∂r
∂v ×
∂r
∂u =
0 ,
the surface is called a smooth surface. If at a point with surface coordinates (u 0 , v 0 )
this cross product equals the zero vector, the point on the surface is called a singular
point of the surface.
