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388 CHAPTER 12 Vector Integral Calculus

12.5 Surface Integrals

Just as there are integrals of vector fields over curves, there are also integrals of vector fields over
surfaces. We begin with some facts about surfaces.

3
A curve in R is given by coordinate functions of one variable, and may be thought of
as a one-dimensional object (such as a thin wire). A surface is defined by coordinate or
parametric functions of two variables,

x = x(u,v), y = y(u,v), z = z(u,v)
for (u,v) in some specified set in the u,v-plane. We call u and v parameters for the surface.

EXAMPLE 12.15

Figure 12.10 shows part of the surface having coordinate functions
1
2
x = u cos(v), y = u sin(v), z = u sin(2v)
2
in which u and v can be any real numbers. Since z = xy, the surface cuts any plane z = k in a
2
hyperbola xy = k. However, the surface intersects a plane y =±x in a parabola z =±x . For this
reason the surface is called a hyperbolic paraboloid.

Often a surface is defined as a level surface f (x, y, z) = k, with f a given function. For
example
2
2
2
f (x, y, z) = (x − 1) + y + (z + 4) = 16
has the sphere of radius 4 and center (1,0,−4) as its graph.
We may also express a surface as a locus of points satisfying an equation z = f (x, y) or

2
2
y =h(x, z) or x =w(y, z). Figure 12.11 shows part of the graph of z =6sin(x − y)/ 1 + x + y .

4
4
2
0 2
0 0
1 1 2 3 –2
2 –2 –4
3 2
–4 4
–2

–4
FIGURE 12.10 The surface x =
u cos(v),y = u sin(v),z = FIGURE 12.11 The surface z = 6sin(x − y)/
2
2
(u /2) sin(2v) in Example 12.15. 1 + x + y .
2

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October 14, 2010 14:53 THM/NEIL Page-388 27410_12_ch12_p367-424