r(u)=⟨a(3u−u3), 3au2, a(3u+u3)⟩
r′(u)=⟨3a(1−u2), 6au, 3a(1+u2)⟩
r′′(u)=⟨−6au, 6a, 6au⟩
r′′′(u)=⟨−6a, 0, 6a⟩
r′×r′′=∣∣i3a(1−u2)−6auj6au6ak3a(1+u2)6au∣∣=18a2(u2−1)i−36a2uj+18a2(u2+1)k
∣r′×r′′∣=18a2(u2−1)2+(−2u)2+(u2+1)2=182a2(u2+1)
∣r′∣=3a(1−u2)2+(2u)2+(1+u2)2=32a(u2+1)
(r′×r′′)⋅r′′′=∣∣3a(1−u2)−6au−6a6au6a03a(1+u2)6au6a∣∣=216a3
κ=∣r′∣3∣r′×r′′∣=542a3(u2+1)3182a2(u2+1)=3a(u2+1)21
τ=∣r′×r′′∣2(r′×r′′)⋅r′′′=648a4(u2+1)2216a3=3a(u2+1)21
So, κ=τ .
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