The curvature at a point of a curve is defined as where is the angle of inclination of

Chapter 10, Problem 69

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The curvature at a point of a curve is defined as where is the angle of inclination of the tangent line at , 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 direction of the curve at and will be studied in greater detail in Chapter 13. (a) For a parametric curve , , derive the Formula where the dots indicate derivatives with respect to , so . [Hint: Use and Formula 2 to find . Then use the Chain Rule to find .] (b) By regarding a curve as the parametric curve , , with parameter , show that the formula in part (a) becomes