Page 409 - Advanced engineering mathematics

12.5 Surface Integrals 389

We often write a position vector

R(u,v) = x(u,v)i + y(u,v)j + z(u,v)k

for a surface. R(u,v) can be thought of as an arrow from the origin to the point
(x(u,v), y(u,v), z(u,v)) on the surface.
Although a surface is different from its graph (the surface is a triple of coordinate functions,
3
the graph is a geometric locus in R ), often we will informally identify the surface with its graph,
just as we sometimes identify a curve with its graph.
A surface is simple if it does not fold over and intersect itself. This means that R(u 1 ,v 1 ) =
R(u 2 ,v 2 ) can occur only when u 1 = u 2 and v 1 = v 2 .

12.5.1 Normal Vector to a Surface
We would like to define a normal vector to a surface at a point. Previously this was done for level
surfaces.
Let be a surface with coordinate functions x(u,v), y(u,v), z(u,v).Let P 0 be a point on
corresponding to u = u 0 ,v = v 0 .
If we fix v = v 0 we can define the curve u on , having coordinate functions

x = x(u,v 0 ), y = y(u,v 0 ), z = z(u,v 0 ).

The tangent vector to this curve at P 0 is

∂x ∂y ∂z
= (u 0 ,v 0 )i + (u 0 ,v 0 )j + (u 0 ,v 0 )k.
T u 0
∂u ∂u ∂u
Similarly, we can fix u = u 0 and form the curve v on the surface. The tangent to this curve
at P 0 is
∂x ∂y ∂z
= (u 0 ,v 0 )i + (u 0 ,v 0 )j + (u 0 ,v 0 )k.
T v 0
∂v ∂v ∂v
These two curves and tangent vectors are shown in Figure 12.12.

z

T u
0
Σ u
T v
0
Σ v y

Σ

x
FIGURE 12.12 Curves u and v and
.
tangent vectors T u 0 and T v 0

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October 14, 2010 14:53 THM/NEIL Page-389 27410_12_ch12_p367-424