Page 388 - Advanced engineering mathematics

368 CHAPTER 12 Vector Integral Calculus

z

y

x
FIGURE 12.1 A nonsimple
curve.

If C is smooth and we let R(t) = x(t)i + y(t)i + z(t)k be the position function for C, then

R (t) is continuous tangent vector to C. Smoothness of C means that the curve has a continuous
tangent vector as we move along it.
A curve is simple if it does not intersect itself at different times. The curve whose graph is
shown in Figure 12.1 is not simple. A closed curve has the same initial terminal points, but is
still called simple if it does not pass through any other point more than once.
We must be careful to distinguish between a curve and its graph, although informally we
often use these terms interchangeably. The graph is a drawing, while the curve carries with it a
sense of orientation from an initial to a terminal point. The graph of a curve does not carry all of
this information.

EXAMPLE 12.1

Let C have coordinate functions
x = 4cos(t), y = 4sin(t), z = 9 for 0 ≤ t ≤ 2π.
The graph of C is a circle of radius 4 about the origin in the plane z = 9. C is simple, closed and
smooth.
Let K be given by
x = 4cos(t), y = 4sin(t), z = 9 for 0 ≤ t ≤ 4π.
The graph of K is the same as the graph of K, except that a particle traversing K goes around
this circle twice. K is closed and smooth but not simple. This information is not clear from the
graph alone.
Let L be the curve given by
x(t) = 4cos(t), y = 4sin(t), z = 9 for 0 ≤ t ≤ 3π.
The graph of L is again the circle of radius 4 about the origin in the plane z = 9. L is smooth
and not simple, but L is also not closed, since the initial point is (4,0,9) and the terminal point
is (−4,0,9). A particle moving along L traverses the complete circle from (4,0,9) to (4,0,9)
and then continues on to (−4,0,9), where it stops. Again, this behavior is not clear from the
graph.

We are now ready to define the line integral, which is an integral over a curve.

Suppose C is a smooth curve with coordinate functions x = x(t), y = y(t), z = z(t) for
a ≤ t ≤ b.Let f, g and h be continuous at least at points on the graph of C. Then the line

integral fdx + gdy + hdz is defined by
C

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October 14, 2010 14:53 THM/NEIL Page-368 27410_12_ch12_p367-424