Proof of the Envelope theorem for parametric functions in the Cartesian PlaneGiven a family of parametric functions in the Cartesian Plane (f(t,k); g(t,k)), their envelope is found by solving the equation: ∂ f ∂ t ∂ g ∂ k − ∂ f ∂ k ∂ g ∂ t = 0How do I prove the above statement?

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Proof of the Envelope theorem for parametric functions in the Cartesian PlaneGiven a family of parametric functions in the Cartesian Plane (f(t,k); g(t,k)), their envelope is found by solving the equation:            ∂      f              ∂      t                  ∂      g              ∂      k        −            ∂      f              ∂      k                  ∂      g              ∂      t        =  0How do I prove the above statement?

lorienoldf7 · Accepted Answer

Step 1In the first place, you should never use the letter k for a continuous variable.The condition you give concerns the Jacobian       J    Φ   of the map  Φ  :    (  t  ,  c  )  ↦            (        f  (  t  ,  c  )  ,  g  (  t  ,  c  )            )           .Step 2If       J    Φ    (      t    0    ,      c    0    )  ≠  0 the map   Φ maps a window with center   (      t    0    ,      c    0    ) diffeomorphically onto a neighborhood of             (        f  (      t    0    ,      c    0    )  ,  g  (      t    0    ,      c    0    )            )      , and this implies that the family of arcs      γ    c    :    t  ↦            (        f  (  t  ,  c  )  ,  g  (  t  ,  c  )            )          (      t    0    −  h  &amp;lt;  t  &amp;lt;      t    0    +  h  )with       c    0    −  δ  &amp;lt;  c  &amp;lt;      c    0    +  δ looks like a family of parallel segments. It follows that the point       p    0    :=            (        f  (      t    0    ,      c    0    )  ,  g  (      t    0    ,      c    0    )            )       cannot belong to the envelope of the given family of curves. Therefore your condition is necessary for envelope points. Sufficiency is another matter.

Proof of the Envelope theorem for parametric functions in the Cartesian Plane Given a family of par

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