Figure 8-13. Surface Sbounded by a simple closed curve Con the surface.
Let z=z(x, y )define the surface S and consider the projections of the surface
S and the curve C onto the plane z= 0 as illustrated in figure 8-13. Call these
projections Rand Cp. The unit normal to the surface has been shown to be of the
form
ˆen=
−
∂z
∂x ˆe^1 −
∂z
√ ∂y ˆe^2 +ˆe^3
1 +
(
∂z
∂x
) 2
+
(
∂z
∂y
) 2 (8 .45)
Consequently, one finds
eˆn·ˆe 2 =
−∂y∂z
√
1 + ( ∂z∂x )^2 + ( ∂z∂y )^2
, ˆen·ˆe 3 =
1
√
1 + (∂z∂x )^2 + ( ∂z∂y )^2
(8 .46))
The element of surface area can be expressed as
dS =
√
1 +
(
∂z
∂x
) 2
+
(
∂z
∂y
) 2
dx dy
and consequently the integral on the left side of equation (8.44) can be simplified to
the form
∫∫
S
(
∂F 1
∂z ˆe^2 ·ˆen−
∂F 1
∂y ˆe^3 ·eˆn
)
dS =
∫∫
S
(
−
∂F 1
∂z
∂z
∂y −
∂F 1
∂y
)
dx dy (8 .47)
