DIFFERENTIAL GEOMETRY - FINAL EXAM - SPRING 2006

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An object is moving in the plane along the curve y = cos(2x) at a constantspeed of 3 ft/sec in the direction of increasing x.

1. Find ||~aT || and ||~aN || at the point (x, cos(2x)).2. Find velocity and acceleration vectors at the point (π

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2).

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An object is moving along a curve in 3-space. At a certain instant, t = t0, theposition vector is ~α(t0) = (2,−1, 3), the velocity vector is ~α′(t0) = (1, 2,−2),the acceleration vector is ~α′′(t0) = (1, 1, 0), and ~α′′′(t0) = (1, 0,−3). At timet = t0 find

a) point in 3-space where the object is located

b) speed of the object

c) unit tangent vector ~T

d) principal normal vector ~N

e) unit binormal vector ~B

f) equation of the rectifying plane

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g) tangential acceleration ~aT

h) normal acceleration ~aN

i) curvature κ

j) torsion τ

k) is the object speeding up or slowing down at time t = t0? Why?

Note: the rectifying plane is formed by ~T and ~B. In this problem we onlyknow ~r, ~v and ~a at one instant t = t0. Do not attempt to find ~r, ~vand ~a as functions of time!

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Let X : R → M be a 2-segment defined on the rectangle R : 0 ≤ u ≤ π2,

0 ≤ v ≤ π. If φ is the 1-form on M such that

φ(Xu) = u cos(v) and φ(Xv) = v sin(u),

verify Stokes’s theorem by computing∫ ∫

X dφ and∫∂X φ separately.

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Consider the surface M with the equation z = (x + y)2. Show that vectors~u1 = (1, 0, 0) and ~u2 = (0, 1, 0) are tangent to M at the origin. ExpressS(a ~u1 + b ~u2) in terms of ~u1 and ~u2, and determine the rank of S at theorigin.

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Let ~α be an asymptotic curve in M ⊂ R3 with curvature κ > 0.a) Prove that the binormal ~B of ~α is normal to the surface along ~α, and

deduce that S(~T ) = τ ~N.b) Show that along ~α the surface has Gaussian curvature K = −τ 2.

Hint: If ~v and ~w are linearly independent tangent vectors, then

S(~v)× S(~w) = K~v × ~w

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Graduate students only!!! Suppose that a curve α lies in two surfacesM1 and M2 that make a constant angle along α. Show that α is principal inM1 if and only if it is principal in M2.

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