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)].