In the calculus of plane curves, one learns that the curvature ) of the curve y D y.x/
Chapter 3, Problem 72(choose chapter or problem)
In the calculus of plane curves, one learns that the curvature ) of the curve y D y.x/ at the point .x;y/ is given by ) D jy00.x/j 1Cy0.x/2"3=2 ; and that the curvature of a circle of radius r is ) D 1=r. [See Example 3 in Section 11.6 of Edwards and Penney,Calculus: Early Transcendentals, 7th edition (Upper Saddle River, NJ: Prentice Hall, 2008).] Conversely, substitute 'Dy0 to derive a general solution of the second-order differential equation ry00 D 1C.y0/2"3=2(with r constant) in the form .x !a/2 C.y !b/2 D r2: Thusacircleofradius r (orapartthereof)istheonly plane curve with constant curvature 1=r.
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