辅导案例-147A
Exam 1 Math 147A Summer 2020
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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.
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