課程名稱︰幾何學
課程性質︰系必修
課程教師︰張樹城
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010年11月03日
考試時限（分鐘）：兩小時
是否需發放獎勵金：是
試題 :
1.(a)(10) Let d be the exterior derivative in R^n, show that d^2 = 0
(b)(5) Let ω = M(x,y)dx + N(x,y)dy be a smooth 1-form in R^2. Show that if
ω is exact, then
∂M ∂N
------ = ------
∂y ∂x
(c)(5) Let
-y x
ω = ------------dx + -----------dy
x^2 + y^2 x^2 + y^2
be a 1-form in R^2\{(0,0)}.Show that ω is closed.
(d)(5) As in (c), is ω excat or not? Give the reason!
2. Let
n ∂ n ∂
X = Σ φ ------ , Y = Σ ψ ------
i=1 i ∂xi j=1 j ∂xj
be two vector fields in R^n
(a)(5) Check ▽ Y and ▽ X.
X Y
(b)(10) Show that [X,Y] = ▽ Y - ▽ X.
X Y
3.(10) Let α be a unit speed curve with κ>0, τ≠0.
Show that α lies on a sphere of radius r if and only if
K^2 + (K'P)^2 = r^2. Here K = 1/κ, P = 1/τ.
4. Consider the following spherical coordinate in R^3:
x = ρcosψcosθ , y = ρsinψcosθ , z = ρcosψ
(a)(5) Check the spherical frame field
∂ 1 ∂ 1 ∂
E1 = ------ , E2 = ------- ---- , E3 = ---- ----.
∂ρ ρsinψ ∂θ ρ ∂ψ
(b)(5) Find the dual coframe {θ1,θ2,θ3}.
(c)(10) Find the connection 1-forms ωi,j.
(d)(5) Compute ▽ E3 , ▽ E3.
E1 E2
(e)(10) Show that the Cartan's structure equations.
5. Let β(s) be a unit speed curve with κ > 0 , torsion τ in R^3.
(a)(5) Find the Frenet frame T,N,B.
(b)(10) Show that
[ T' ] [ ０ κ ０ ][ T ]
[ N' ] = [ -κ ０ τ ][ N ].
[ B' ] [ ０ -τ ０ ][ B ]
6.(10) Consider the mapping x : D →R^3. Here D is open in R^2.
Set E = xu ‧ xu , F = xu ‧ xv , G = xv ‧ xv.
(a) Show that || xu × xv ||^2 = EG -F^2.
(b) x is regular if and only if EG - F^2 ≠ 0 on x(D).
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