Chapter 12.5, Problem 39ES

Use the formula in Exercise 23 (a) to show that for a curve in polar coordinates described by $r=f\left(\theta \right)$ the curvature is

$\kappa \left(\theta \right)=\frac{\left|r^2+2{\left(\frac{dr}{d\theta }\right)}^2-r\frac{d^{2}r}{d{\theta }^2}\right|}{{\left[r^2+{\left(\frac{dr}{d\theta }\right)}^2\right]}^{3/2}}$