Envelope of a parametric family of functions

[Icaro Lorran]

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 $J_{\phi}(t,s)=0$?

 

[WWGD]

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 $J_{\phi}(t,s)=0$?

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).