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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