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346 CHAPTER 11 Vector Differential Calculus

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80
–1
60
–0.5 40
20
0
0
20
0.5
40
1
60
80

100

FIGURE 11.1 Graph of the curve of Example 11.1.

F(t + h) – F(t )
0
0
F (t )
0
(f(t ), g(t ), h(t ))
0
0
0
z
F(t )
0
F(t + h)
0
y

x
FIGURE 11.2 F (t 0 ) as a tangent vector.

To give an interpretation to the vector F (t 0 ), look at the limit of the difference quotient:

F(t 0 + h) − F(t 0 )
F (t 0 ) = lim

h→0 h

x(t 0 + h) − x(t 0 ) y(t 0 + h) − y(t 0 )
= lim i + lim j
h→0 h h→0 h

z(t 0 + h) − z(t 0 )
+ lim k
h→0 h

= x (t 0 )i + y (t 0 )j + z (t 0 )k.
Figure 11.2 shows the vectors F(t 0 + h), F(t 0 ) and F(t 0 + h)−F(t 0 ), using the parallelogram
law. As h is chosen smaller, the tip of the vector F(t 0 + h) − F(t 0 ) slides along C toward F(t 0 ),
and (1/h)[F(t 0 + h) − F(t 0 )] moves into the position of the tangent vector to C at the point
( f (t 0 ), g(t 0 ),h(t 0 )). In calculus, the derivative of a function gives the slope of the tangent to the
graph at a point. In vector calculus, the derivative of the position vector of a curve gives the
tangent vector to the curve at a point.

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October 15, 2010 16:11 THM/NEIL Page-346 27410_11_ch11_p343-366