Chapter 12.5, Problem 23ES

(a) Use Formula (3) to show that in 2-space the curvature of a smooth parametric curve

is $\begin{array}{l}x=x\left(t\right),y=y\left(t\right)\\ \kappa \left(t\right)=\frac{\left|x^{\prime }{y}^{″}-y^{\prime }{x}^{″}\right|}{{\left({x^{\prime }}^2+{y^{\prime }}^2\right)}^{3/2}}\end{array}$ where primes denote differentiation with respect to t.

(b) Use the result in part (a) to show that in 2-space the curvature of the plane curve given by

$y=f\left(x\right)$ is $\kappa \left(x\right)=\frac{\left|d^{2}y/d{x}^2\right|}{{\left[1+{\left(dy/dx\right)}^2\right]}^{3/2}}$