Concordia University February 16, 2011
MATH 380
Differential Geometry
Midterm
Instructions: No cell phones and no calculators are allowed during the examination. You
have 1 hour to address the following questions. Please show your work clearly.
(1) Consider
vp= (1,−1,2) a tangent vector of R3at the point p= (0,1,−2),
ϕ=yz dx +y dy −x2dz a 1-form,
f=x2−yz a real-valued function,
V=U1+ 2xyU2−z2U3a vector field,
F:R3→R3, F (u, v, w) = (u−v , u2, u −w) a mapping.
(a) (10 points) Compute vp[f] and ∇vp(fV ).
(b) (10 points) Evaluate ϕ(vp) and dϕ.
(c) (8 points) Describe the image of the y-axis under the map F. Find F⋆(vp).
(d) Extra Credit (6 points): Calculate dϕ(vp,vp) and dϕ(vp,V(p)). Is the mapping
Ffrom part (c) regular? Explain.
(2) (8 points each)
(a) Show that a curve has constant speed if and only if its acceleration is everywhere
orthogonal to its velocity.
(b) Show that the curve α(t) = (cosh t, sinh t, t), t ∈Rhas arc length function s(t) =
√2 sinh t, and find the unit speed reparametrization of α.
(c) Find the Frenet frame, {T, N, B }, of the curve
β(s) = ((1 + s)3/2
3,(1 −s)3/2
3,s
√2),−1< s < 1.
(d) The coordinates x(t) = t−sin t, y(t) = cos t−1, with 0 < t < 2π, describe a curve
in the xy-plane. Find its curvature function.
Some Formulas: sinh t=et−e−t
2,cosh t=et+e−t
2.
k=||α′×α′′||/||α′||3, τ = (α′×α′′)·α′′′ /||α′×α′′||2.
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