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Suppose that F: Rn -> Rn is continuously differentiable everywhere

  1. Apr 24, 2012 #1
    Please help me with this. I dont know even how to start

    Definition: Suppose that F: Rn Rn is continuously differentiable everywhere. A point P∈R^n is called an isolated singularity of F if DF_p is not invertible but DF_y is invertible for all Y≠P in some neighborhood of P.

    a. Let f: R -> R by f(x) = 3 x^4 - 20 x^3. Prove that f has exactly two points which are isolated singularities. Describe what happens at each point.

    b. Let f denote the transformation that takes rectangular coordinates into polar coordinates. Does f have any isolated singularities? If so, identify them and explain what happens at each point?

    c. Write down explicitly a continuously differentiable function F : R2 -> R2 that has an isolated singularity at the origin and no other singularity.
     
  2. jcsd
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