James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

864 Chapter 13 Vector Functions
8 Definition The curvature of a curve is
dT
− Z
ds
Z
where T is the unit tangent vector.
The curvature is easier to compute if it is expressed in terms of the parameter t instead
of s, so we use the Chain Rule (Theorem 13.2.3, Formula 6) to write
dT
− dT ds
dT dTydt
and −
dt ds dt
Z
ds
Z − Z
dsydt
Z
But dsydt − | r9std | from Equation 7, so
9 std − | T9std |
| r9std |
ExamplE 3 Show that the curvature of a circle of radius a is 1ya.
Solution We can take the circle to have center the origin, and then a parametrization
is
rstd − a cos t i 1 a sin t j
Therefore r9std − 2a sin t i 1 a cos t j and | r9std | − a
so Tstd − r9std
| r9std − 2sin t i 1 cos t j
|
and
This gives | T9std |
T9std − 2cos t i 2 sin t j
− 1, so using Formula 9, we have
std − | T9std |
| r9std | − 1 a
■
The result of Example 3 shows that small circles have large curvature and large circles
have small curvature, in accordance with our intuition. We can see directly from the definition
of curvature that the curvature of a straight line is always 0 because the tangent
vec tor is constant.
Although Formula 9 can be used in all cases to compute the curvature, the formula
given by the following theorem is often more convenient to apply.
10 Theorem The curvature of the curve given by the vector function r is
std − | r9std 3 r0std |
| r9std | 3
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