课程名称︰几何学 课程性质︰系必修 课程教师︰张树城 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010年11月03日 考试时限（分钟）：两小时 是否需发放奖励金：是 试题 : 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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