課程名稱︰幾何學
課程性質︰數學系大三必修課
課程教師︰蔡宜洵
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/11/14
考試時限（分鐘）：1:20 ~ 3:20
試題 :
Total points: 105
Notice: any theorems or facts put to use in the process of your proof or
computation must be stated clearly.
s
1 (20 pts in total). Given the parametrized curve (helix) α(s) = (a cos— ,
c
s s 2 2 2
a sin—, b— ), s∈R where a, b, c positive, c = a + b . i)(5 pts) Show that
c c
the parameter s is the arc length. ii)(15 pts) Determine the curvature, torsion
and osculating plane of α.
2 (20 pts in total). The gradient of a differentiable function f : S → R is a
3
differentiable map grad(f) : S → R which assigns to each point p∈S a vector
3
grad(f)∈T (S)⊂R such that < grad(f(p)), v > = df (v) for all v∈T (S).
p p p p
f G - f F f E - f F
u v v u
i)(12 pts) Show that grad(f) = —————— x + —————— x with E, F, G
2 u 2 v
EG - F EG - F
the coefficients of the first fundamental form in a local parametrization
x : U → S. ii)(8 pts) If you let p∈S be fixed and v vary in the unit circle
grad(f)
|v| = 1 in T (S), then df (v) is maximum if and only if v = —————.
p p |grad(f)|
2 2
3 (20 pts in total). Let C be the unit circle: x + y = 1 and a differentiable
family of straight lines {L } be defined in such a way that for each p∈C,
p p∈C
π
L ∋p is constructed to make an angle —— with the normal n (to C at p) and
p 6 p
at p = (1, 0), L has a positive slope. i)(8 pts) Put the family as constructed
p
above into the form given explicitly by a one parameter family of equations.
ii)(12 pts) Find the envelope (caustic curve) of this family.
2 2 2
x y z
4 (20 pts in total). S : —— + —— + —— = 1. i)(15 pts) Show that the Gauss
2 2 2
a b c
2
map N : S → S is one to one, onto and differentiable. ii)(5 pts) Let K = K(p)
be the Gaussian curvature of S (at p) and dσ be the surface area form of S,
2
i.e. dσ = √(EG - F ) du dv in local parameters. Show/Explain that ∫K dσ
S
= 4π.
5 (10 pts in total). Given a vector field w : q = (x,y) → w(q) = (1-x,1+y)
on plane. i)(5 pts) For p = (A,B) find explicitly the trajectory through p.
ii)(5 pts) Find an explicit first integral of the above vector field w.
6 (15 pts in total). For v∈T (S) where S is a regular surface, let H be
p v,α
π
either of the two planes containing p, v and making an angle 0 < α≦ ——
2
with T (S). Let e , e ∈T (S) be principle directions with principle
p 1 2 p
curvatures k ≧ k ＞ 0. Given v = cosθe + sinθe (θ≠0) let the section C
1 2 1 2 θ
be H ∩ S and k be the curvature (at p) of C . Denote by k (C) the normal
v,θ θ θ n
curvature of a curve C (at p). i)(5 pts) In the above notation given two
sections C , C (θ≠θ) with curvature k , k respectively. In terms of
θ θ 1 2 θ θ
1 2 1 2
k , k , θ, θ, find k (C ), k (C ) and k , k . ii)(5 pts) Given any v≠0,
θ θ 1 2 n θ n θ 1 2
1 2 1 2
v∈T (S) and any M > 0, does there exist a curve C⊂S passing through p, v
p
∈T (C) and having its curvature k (p) > M ? Justify your answer. iii)(5 pts)
p C
Fix a θ≠0 and v≠0 as above. Is it possible to find a curve C∋p in S,
v∈T (C), with the property that it has the osculating plane H while its
p v,θ
curvature k (p) ≠ k ? More generally, do there exist two curves (in S)
C θ
through p, sharing the same tangent line (at p) and having the same
osculating plane (at p) while their curvatures (at p) are different? Justify
your answer.