95年下學期期末考題 1.Determine the compactness,connectness and simply connectness of the following surfaces: (A)the region z<0 in M: z=xy , (B)the torus, (C) F(M), where F is a diffeomorphism of a simply connected and compact surface M onto a surface N. Given explaination for your answers.(Some of then can use graph to explain.) 2.Let M be a sueface in R^3. (A)If η and ξ are 2-forms on M such that η(Xu,Xv) = ξ(Xu,Xv), for every patch X in M. Is η=ξ? Verify your answer. (B)Let ζ be 2-form on R^2. Is ζ=ζ(U1,U2)dudv? Verify your answer. (C)Let X(u,v)=(x(u.v),y(u,v),z(u,v)) be a 2-segment in M. Find the pullback of the 2-form f(x,y,z)dzdx on M under X , where f is defined on the image of X in M. 3.Let M be a surface in R^3, X:R→M be a 2-segment in M and Φ be a 1-form on M, Prove the stoke THM :∫∫dΦ = ∫Φ , X aX 4.(A)Is the line interal of any exact 1-form path independent? Verify your answer. (B)Is the line interal of any closed 1-form path independent? Verify your answer. 5.Prove or disprove that given surfaces (A)M:x^4 + y^2 + z^2 = 1 ,and (B)the Mobius band B are orientable. 6.If Σ is the sphere ||P||=r, the mapping A:Σ→Σ such that A(P)=-p is called the antipodal map of Σ. Prove that A is a diffeomorphism and that A*(Vp)=(-v) -p (註: * 和 -p 是下標) 7.Let X be a patch in an surface M. Show that for p=x(u,v) , Xu(u,v) × Xv(u,v) is normal to Tp(M). -- 這輩子.. 最大的憾事... 莫過於... 　　　『 沒在 ５５６６板 被 "劣文" 被 "水桶" ！！！ 』　 --

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