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Conjugate of every variable is denoted

  1. Nov 26, 2008 #1
    Hi I have received PM so I would like to post this question for scientific debate

    here conjugate of every variable is denoted by _.
    1.define f_:R3->R3 by f_(x_)=y_, where y1=x1+(x2)2+(x3-1)2,

    y2=(x1)2+x2+((x3)2-3.x3), y3=(x1)3+(x2)2+x3.prove that f_ is locally

    invertible about x_=(0,0,1).

    2.let x_(theta,v)= (cos theata,sin theta,0)+v(sin 1/2 theta.cos theta,

    sin1/2theata.sin theta,cos1/2 theta) with -pi<theata<pi,-1/2<v<1/2.

    compute n_(theta,0) and show that

    lim theta->-pi n_(theta,0)=-lim theta->pi n_(theta,0).

    while lim theta->-pi x_(theta,0)=-lim theta->pi x_(theta,0).

    [this is called the mobius band].give anothe coordinate patch so that

    theta=+\-pi is included, thus making it to a surface [note that it is

    impossible to make a choice of unit normal at each point of this surface

    in a continuous fashion].

    3.let M be the surface x_(u1,u2)=(u2cosu1,u2sinu1,pu1),where p is a

    non zero constant.show that M is a minimal .[Mis called a hellicoid.]

    4.if M1(resp.M2) is a C infinity n1-manifold (resp.n2-manifold),prove

    M1xM2 is a C infinity n-manifold for n=n1+n2.

    [hint.if (Ui,Qi) is a chart about mi belongs to Mi,than that make

    (U1xU2,Q1xQ2) achart about (m1,m2)].
    Last edited by a moderator: Nov 26, 2008
  2. jcsd
  3. Nov 26, 2008 #2


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    Homework Helper
    Gold Member

    Re: Advance

    Is something wrong with normal 12pt black text?!...Your post is an eyesore to try and read.

    Have you made any attempt at these problems? Is this homework?
  4. Nov 26, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    Re: Advance

    "Scientific debate"? Why should anyone debate something as simple as this?

    The only debate is what do you mean by "conjugate"?
  5. Nov 26, 2008 #4
    Re: Advance

    Hi I got this question from private message therefore I posted it if anyone feel it is homework do not answer on it because I don't know if it homework or not even it is as simple as this
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