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" ！！！ 』
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.228.152.112
※ 編輯: shihweng 來自: 61.228.151.237 (05/09 02:06)