Surface Area
The position vector
r =r (u, v ) = x(u, v )ˆe 1 +y(u, v )ˆe 2 +z(u, v )ˆe 3 α≤u≤β, γ ≤v≤δ (7 .51)
defines a surface in terms of two parameters uand v. The family of curves r (u, v 0 ),
with v 0 taking on selected constant values, defines a set of coordinate curves on the
surface. Similarly, the family of curves r (u 0 , v ), with u 0 taking on selected constant
values, defines another set of coordinate curves. The vector ∂r∂u is a tangent vector to
the coordinate curve r (u, v 0 )and the vector ∂r∂v is a tangent vector to the coordinate
curve r (u 0 , v ). If at every common point of intersection of the coordinate curves
r (u 0 , v )and r (u, v 0 )one finds that ∂r∂v ·∂r∂v
(u 0 ,v 0 )
= 0 , then the coordinate curves are
said to form an orthogonal net on the surface.
The vector
dr =
∂r
∂u du +
∂r
∂v dv (7 .52)
lies in the tangent plane to the point r (u, v ) on the surface and one can say that
the vector element dr defines a parallelogram with vector sides ∂u∂r du and ∂r∂v dv as
illustrated in the figure 7-13.
Figure 7-13. Defining an element of area on a surface.
Define the element of surface area dS on a given surface as the area of the elemental
parallelogram formed using the vector components of dr. Recall that the magnitude
