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Critical points of maps f:R^n->R^m , m>1

  1. Sep 26, 2010 #1
    Critical points of maps f:R^n-->R^m , m>1

    Hi, Everyone:

    I am having trouble figuring out what the critical points would be for any such map

    f:R^n -->R^m , m>1.

    Problem is that I know sufficient conditions for differentiability (partials exist and

    are continuous.) but not necessary ones. So our differential is given by an

    mxn matrix. In the case of n=m, we just want the Jacobian matrix to be invertible,

    but I am not clear on how to deal with this issue when n=/m.

    Any Hints, Please.?

    Thanks.
     
  2. jcsd
  3. Sep 28, 2010 #2
    Re: Critical points of maps f:R^n-->R^m , m>1

    In the case m=n, why do you say that the jacobian must be invertible for the function to be differentiable? Consider the null function from R^n to itself. This is differentiable, but the jacobian is not invertible. I think you confuse the concepts of differentiable functions and diffeomorphisms (invertible differentiable functions). Or maybe I misunderstood your question.
     
  4. Oct 2, 2010 #3
    Re: Critical points of maps f:R^n-->R^m , m>1

    Thanks, Mugver:

    What do you mean by the null function.?

    Actually, the condition of the Jacobian being invertible is, I think, sufficient,

    but not neccessary: I think we assume when we have a Jacobian, that

    the partials Delf/Delx_i are continuous. Then we have that the partials

    exist and they are continuous. Then f is differentiable.
     
  5. Oct 2, 2010 #4
    Re: Critical points of maps f:R^n-->R^m , m>1

    I mean the function from R^m to R^n defined by f(x) = 0.

    The rank of the matrix formed by the derivatives of the function has nothing to do with the differentiability of the function. Maybe you should state more precisely your question... :)
     
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