
Applications of the Derivative to Geometry 6-41
ψθ
ψ
T
X
P
Figure 6.19
Differentiating w.r.t q, we obtain
d
d
dr
d
r
d r
d
r
dr
d
φ
θ
θ
θ
θ
=
−
+
2
2
2
2
2
.
Thus,
d
d
dr
d
r
d r
d
r
dr
d
r r
d r
d
ψ
θ
θ
θ
θ
θ
=
−
+
+
=
−
2
2
2
2
2
2
2
1
22
2
2
2
2+
+
dr
d
r
dr
d
θ
θ
.
We know that the derivative of the arc is given by
ds
d
r
dr
dq q
= +
2
2
1 2/
.
Therefore, the curvature of a curve in polar coordinates is given by
K
d
ds
r r
d r
d
dr
d
r
dr
d
= =
− +
+
ψ
θ
θ
θ
2
2
2
2
2
2
2
3 2/
.