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From: dushya on 17 Mar 2010 18:07 hi can anyone please suggest me some theorems/results which can be used in proving or disproving following statements --- 1. given a differentiable curve x:[a,b] --> Rn (n-dimensional euclidean space). and a real valued function f defined along the curve x such that fox is differentiable. then there exist an open set U containing curve x and a differentiable real valued function g on U such that g restricted to curve x is equal to f. 2. let D be the closed disk { (x1,x2) : square (x1) +square (x2) <=1 }. consider cylinder S= Dx[0,1] and let y:[0,1] -->R3 be a differentiable curve. then there exist a bijective differentiable map T from S into R3 (euclidean 3-space) such that under T axis of cylinder S is mapped into curve y and each cross section of S is mapped into some plane perpendicular to curve.
From: Robert Israel on 18 Mar 2010 02:39 dushya <sehrawat.dushyant(a)gmail.com> writes: > hi can anyone please suggest me some theorems/results which can be > used in proving or disproving following statements --- > > 1. given a differentiable curve x:[a,b] --> Rn (n-dimensional > euclidean space). and a real valued function f defined along the curve > x such that fox is differentiable. then there exist an open set U > containing curve x and a differentiable real valued function g on U > such that g restricted to curve x is equal to f. Consider n=1, x(t) = t^3, f(x(t)) = t. You must have g(x) = x^(1/3), but that is not differentiable at 0. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: dushya on 18 Mar 2010 21:16
On Mar 17, 11:39 pm, Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote: > dushya <sehrawat.dushy...(a)gmail.com> writes: > > hi can anyone please suggest me some theorems/results which can be > > used in proving or disproving following statements --- > > > 1. given a differentiable curve x:[a,b] --> Rn (n-dimensional > > euclidean space). and a real valued function f defined along the curve > > x such that fox is differentiable. then there exist an open set U > > containing curve x and a differentiable real valued function g on U > > such that g restricted to curve x is equal to f. > > Consider n=1, x(t) = t^3, f(x(t)) = t. You must have g(x) = x^(1/3), but that > is not differentiable at 0. > -- > Robert Israel isr...(a)math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada hey thanks:-) but can there be any conditons on x and/or on f under which statement 1 holds. i read about a theorem (Tietze Extension Theorem ) in some tpology book. according to this theorem,in case of a normal space any continous map on a closed subset can be extended continously to the whole space. is there any anlog of it for differentiable functions? also can you suggest something about statement 2. |