Figure 7-19. Element of surface area and element parallelogram.
In the limit as ∆xand ∆ytend toward zero, ∆Rapproaches ∆Sand one can define
dR =dS , where the element of area dR lies in the tangent plane to the surface at the
point (x, y, z).In the limit as ∆xand ∆yapproach zero, the element of area is defined
as the area of the elemental parallelogram defined by the vector dr and illustrated in
figure 7-19(b). The area of this elemental parallelogram can be calculated from the
cross product relation
(∂r
∂x dx)
×(∂r
∂y dy)
=∣∣
∣∣
∣∣eˆ 1 ˆe 2 ˆe 3
dx 0 ∂z∂x dx
0 dy ∂z∂y dy∣∣
∣∣
∣∣=(
−∂z
∂x ˆe^1 −∂z
∂y ˆe^2 +ˆe^3)
dx dy. (7 .91)The area of the elemental parallelogram is the magnitude of the above cross product,
and can be expressed
dS =dR =√
1 +(
∂z
∂x) 2
+(
∂z
∂y) 2
dx dy. (7 .92)