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# Curves in the plane. For a unit-speed curve b(s) = (x(s), y(s)) in R2, the unit tangent is T = b = (x, y) as usual, but the unit normal N is defined...

Curves in the plane. For a unit-speed curve b(s) = (x(s), y(s)) in R2, theunit tangent is T = b¢ = (x¢, y¢) as usual, but the unit normal N is defined byrotating T through +90°, so N = (-y¢, x¢). Thus T¢ and N are collinear, andthe plane curvature k˜ of b is defined by the Frenet equation T¢ = k˜N.(a) Prove that k˜ = T¢ • N and N¢ = -k˜T.