課程名稱︰幾何學
課程性質︰必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰11/11/2009
考試時限（分鐘）：兩小時三十分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
You may work on each part separately by assuming the
result of other parts.
1.(15) Consider the Helix α: R → R^3 given by α(t)=(acost,asint,bt)
(a) Parametrize the Helix b its arc length s.
(b) Show that both κ(s) and τ(s) are constant in s.
(c) Determine all spae curves with constant κ and τ .
2.(15) Let α: I→ R^3 be a regular smooth curve parametrized by its
arc length such that the Frenet frame {T,N,B} is well-defined on it.
(a) Derive the local canonical form of α near he point α(0).
(b) Let π:R^3 → E be the orthogonal projection onto the osculating
plane E at α(0) spanned by T(0) and N(0). Show that
__ __
the curvature κ(0) of the plane curve α := π。α :I→E ～R^2
at s=0 equals κ(0).
3.(20) Let α: [0,L]→ R^3 be a regular smooth curve parametrized by its
arc length such that the Frenet frame is well-defined on it.
(a) Let S , a subset of R^3, be te tubular surface along α with a fixed
radius r > 0 . Find a parametrization of S using the Frenet frame.
(b) Show that S is a regular surface if r is small enough.
(c) Compute the first fundamental form and the area of S.
4.(20) Regular surfaces defined by level sets:
(a) Let S = F^(-1)(a) be the level set of a smooth function F:R^3 → R
with a (in R) , a regular value.
Prove in detail that S is a regular value.
(b) Show that T_p(S) is the plane orthogonal to the vector ▽F(p).
(c) Consider three surfaces S_1, S_2, S_3 defined by x^2+y^2+z^2 = ax ,
x^2+y^2+z^2 = by ,x^2+y^2+z^2 = cz respectively where a,b,c≠0.
Show that they are all regular surfaces. Moreover,
if p is in S_1∩S_2∩S_3, show that T_p(S_1), T_p(S_2) , T_p(S_3)
intersect each other orthogonally.
5.(20) Let S be a regular surface in R^3 with N : S → S^2 the Gauss map.
Consider a local parametrization x : U (in R^2) → S with coordinates
(u,v) in U .
(a) Show that the matrix representing dN_p with respect to the basis
x_1, x_2 is given by
┌ ┐
1 │fF-eG gF-fG│
──── │ │
EG-F^2 │eF-fE fF-gE│
└ ┘
(b) Find the differential equation for a curve α(t) = x(u(t) , v(t))
on S to be a line of curvature and show that the coorinate curves
are precisely the lines of curvature if and only if F=0 and f=0.
(c) Compute the two principle curvatures of the Enneper' surface given by
x(u,v) = ( u-(u^3)/3+uv^2 , v-(v^3)/3+vu^2 , u^2-v^2) .
6.(10) Let α(v) be a curve in the xz plane and let S be he surface of
revolution of α in the z-axis. Fot α(v) = ( φ(v) , ψ(v) ) with v
being the arc length, compute K, determine all such S with K≡1.
If moreover S is compact and regular, show that S must be he sphere.
7.(10) Let S , a subset R^3 , be a regular surface with K<0 and α be an
asympototic curve on it with κ≠0. Show that |τ|= √(-K).
8.(10) Use Green's Theorem to prove one of the following theorems.
(1) The change of variable formula on R^2 for C1 functions.
(2) The iso-perimetric inequality.
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