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Maping from R2 to R3 torus, finding max min and saddle points

  1. Apr 18, 2009 #1
    1. The problem statement, all variables and given/known data
    this is from ch9 (functions of several variables)of baby rudin

    a,b real b>a>0 define a mapping f=(f1,f2),f3) of R2 into R3 by
    f1(s,t)=(b+acos(s))cos(t)
    f2(s,t)=(b+acos(s))sin(t)
    f2(s,t)=asin(s)
    I showed that there are exactly 4 points p in K=image(f) such that

    gradf1(f-1(p))=0

    I am having trouble finding which of these points are local min, max, and saddle points.


    3. The attempt at a solution

    The points in question correspond to the touple (s,t) when s=(pi)k and t=(pi)j k,j integers.

    the points being +/-b +/-a

    Okay great, I do not have any developed criteria for the second derivative test, I have no Hessin matrix to work with. I can develop that in my problem, but that seems rather complicated, I have been trying for a while now, and am having trouble.
     
  2. jcsd
  3. Apr 18, 2009 #2
    after applying the second derivative test to each component of f I find that at each point of the second derivative of f2 and f3 that the value is zero. So I have zeros at all point of the partial derivatives which is bad because I should have 2 saddle points 1 max and one min. Even more confusing is that in f1 I get two min and two max, no saddle points...
    Any ideas
     
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