# Conjugate of every variable is denoted

1. Nov 26, 2008

### memomath

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

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

Last edited by a moderator: Nov 26, 2008
2. Nov 26, 2008

### gabbagabbahey

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?

3. Nov 26, 2008

### HallsofIvy

Staff Emeritus

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

The only debate is what do you mean by "conjugate"?

4. Nov 26, 2008