Page 367 - Advanced engineering mathematics

11.1 Vector Functions of One Variable 347

In Example 11.1,

H (t) = 2ti + cos(t)j − 2tk,
2
2
and this vector is tangent to the curve at any point (t ,sin(t),−t ) on the curve. The tangent
vector at (0,0,0) is H (0) = j, as we can visualize from Figure 11.1.

The length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤t ≤b is
b

2
2
2

length = (x (t)) + (y (t)) + (z (t)) dt.
a
In vector notation, this is
b

length = F (t) dt.
a
The length of a curve is the integral (over the defining interval) of the length of the tangent vector
to the curve, assuming differentiability at each t.
Now imagine starting at (x(a), y(a), z(a)) at time t = a and moving along the curve, reach-
ing the point (x(t), y(t), z(t)) at time t.Let s(t) be the distance along C from the starting point
to this point (Figure 11.3). Then
t

s(t) = F (ξ) dξ.
a
This function measures length along C and is strictly increasing, hence it has an inverse. At least
in theory, we can solve for t = t(s), writing the parameter t in terms of arc length along C.We
can substitute this function into the position function to obtain
G(s) = F(t(s)).
G is also a position vector for C, except now the variable is s and s varies from 0 to L, the length

of C. Therefore, G (s) is also a tangent vector to C. We claim that this tangent vector in terms of
arc length is always a unit vector. To see this, observe from the fundamental theorem of calculus
that

s (t) = F (t) .

C

z

(x(t), y(t), z(t))

s(t)

(x(a), y(a), z(a))
y

x

FIGURE 11.3 Distance function along a
curve.

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October 15, 2010 16:11 THM/NEIL Page-347 27410_11_ch11_p343-366