Chapter 10.2, Problem 69E

The curvature at a point P of a curve is defined as

$\kappa =\left|\frac{d\varphi }{ds}\right|$

Where $\varphi$ 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 $\varphi$ 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 $x=x\left(t\right),y=y\left(t\right),$ derive the formula

$\kappa =\frac{\left|\stackrel{\dot{}}{x}\ddot{y}-\ddot{x}\stackrel{\dot{}}{y}\right|}{{\left[{\stackrel{\dot{}}{x}}^2+{\stackrel{\dot{}}{y}}^2\right]}^{3/2}}$

where the dots indicate derivatives with respect to t so $\stackrel{\dot{}}{x}=dx/dt$. [Hint: Use $\varphi ={\tan }^{-1}\left(dy/dx\right)$ and Formula 2 to find $d\varphi /dt$. Then use the Chain Rule to find $d\varphi /ds$.]

(b) By regarding a curve $y=f\left(x\right)$ as the parametric curve $x=x,y=f\left(x\right)$, with parameter x, show that the formula in part (a) becomes

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