The curvature at a point P of a curve is defined as
κ
=
|
d
ϕ
d
s
|
Where
ϕ
is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the rate of change of
ϕ
with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 13.
(a) For a parametric curve
x
=
x
(
t
)
,
y
=
y
(
t
)
,
derive the formula
κ
=
|
x
˙
y
¨
−
x
¨
y
˙
|
[
x
˙
2
+
y
˙
2
]
3
/
2
where the dots indicate derivatives with respect to t so
x
˙
=
d
x
/
d
t
. [Hint: Use
ϕ
=
tan
−
1
(
d
y
/
d
x
)
and Formula 2 to find
d
ϕ
/
d
t
. Then use the Chain Rule to find
d
ϕ
/
d
s
.]
(b) By regarding a curve
y
=
f
(
x
)
as the parametric curve
x
=
x
,
y
=
f
(
x
)
, with parameter x, show that the formula in part (a) becomes
κ
=
|
d
2
y
/
d
x
2
|
[
1
+
(
d
y
/
d
x
)
2
]
3
/
2