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Envelope of a parametric family of functions

  1. Jan 15, 2016 #1
    Consider the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##, a point belonging to the envelope of this map satisfy the condition ##J_{\phi}(t,s)=0##. What is the role of the Jacobian in maps like these and why points in the envelope have to satisfy ##J_{\phi}(t,s)=0##?
     
  2. jcsd
  3. Jan 15, 2016 #2

    WWGD

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    I cannot be sure without knowing the full context, but usually if you set the equation to 0 , under mild conditions the set of points satisfying the equation is a (sub)manifold (re the inverse function theorem/implicit function theorem, which uses the invertibility of the Jacobian matrix).
     
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