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

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 with respect to arc length. It can be regarded as a measure of the rate of change of Φ 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

Where the dots indicate derivatives with respect to t, so x = dx/dt.
(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

Transcribed Image Text:

ds ху-ху! dy/dx) ]32 0