Concordia University February 13, 2008
Differential Geometry
Midterm
(1) Consider
vp= (−2,1,−1) a tangent vector of R3at the point p= (1,0,2),
ϕ=y2dx −x2dz a 1-form,
f= (1 + x)yz a real-valued function,
V=xU1+ 2yU2−xy2U3a vector field.
(a) Compute vp[f] and V(p)[f].
(b) Compute V[V[f]].
(c) Evaluate ϕ(vp) and ϕ(V).
(d) Simplify ϕ∧df.
(e) Compute ∇vpVand ∇f(p)vpV.
(2) Let α(t) = ((2t
3+1
2)3
2
,1
2(1
2−2t
3)3
2
,√3
2(1
2−2t
3)3
2), t ∈(−3
4,3
4)be a curve
in R3.
(a) Find the Frenet frame field {T, N, B }at α(t).
(b) Find the curvature function along the curve.
(c) Find the torsion function along the curve and use it to determine if the curve is
planar.
(d) Find a formula for the segment Γ connecting α(0) to α(1/2). Show that Γ is
orthogonal to B(0), the binormal vector at α(0).
(e) Consider β(t), t ∈(−3
4,3
4),the spherical image of αand find its curvature
function.
1