Answer to Question #134209 in Differential Geometry | Topology for Promise Omiponle

Question

Find the curvature of the parabolic curve r(t) =<t,t^2>.

Answer 1

r(t) = t , t^2

r^,(t) = 1, 2t

r^,,(t) = 0 , 2

r^, (t)×r^,,(t) = (0,0,2)

|r^,(t)×r^,,(t)| = (0+0+4)^1/2 = 4^1/2

|r^,(t)| = (1+4t²)^1/2

|r^,(t)|³ = (1+4t²)^3/2

Curvature = |r^,(t)×r^,,(t)|/|r^,(t)|³

= [(4)^1/2/(1+4t²)^3/2] ans.