辅导案例-147A

• September 1, 2020

Exam 1 Math 147A Summer 2020 Your name: Your perm number: Scores: 1. 2. 3. 4. Do any three problems. Do not do more than 3 problems. Each problem is worth 34 points. 1. Let α(t) be an arclength-parametrized C3 curve which is contained in a plane, where t ∈ [a, b] for some a < b. Recall that C3 means that α has continuous derivatives up to third order. We create a “parallel” curve by taking β(t) = α(t) + ϵN(t), where ϵ > 0 is a constant. 1) (6 points) Show that β is a regular curve whenever ϵ is small enough, i.e. whenever 0 < ϵ < ϵ0 for some ϵ0 > 0. 2) (28 points) Express the curvature of β in terms of the curvature of α. 2. Consider the regular curve γ(t) = (3t− t3, 3t2, 3t+ t3), t ∈ R, in R3. a) (9 points) Compute the Frenet frame {T,N,B} of γ(t). (You want to be careful with your calculations as the results should be reasonably clean.) b) (25 points) Verify that γ(t) is a generalized helix. 3. Let γ(s) be a unit-speed curve in R3 and {T,N,B} its Frenet frame. Recall that the osculating plane P (s) of γ at a point γ(s) is the plane passing through γ(s) and spanned by the tangent vector T(s) and the principal normal N, i. e. P (s) = {γ(s) + xT(s) + yN(s) : x, y ∈ R}. Suppose that all osculating planes of γ pass through a given point p0. Prove that γ is a planar curve. 4. Let α be a regular curve with κt0) ̸= 0 for some t0. Let β denote the planar curve obtained by projecting α into its osculating plane at p0 = α(t0). Prove that β has the same curvature at p0 (i. e. at the time t0) as α. 1

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