The discussion focuses on the mathematical concept of the envelope of a parametric family of functions represented by the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##. It establishes that points on the envelope must satisfy the condition ##J_{\phi}(t,s)=0##, where ##J_{\phi}## denotes the Jacobian of the map. The Jacobian's role is crucial as it indicates the conditions under which the set of points forms a manifold, based on the inverse function theorem and the properties of the Jacobian matrix.
PREREQUISITESMathematicians, students of calculus and differential geometry, and anyone interested in the applications of Jacobians in multivariable functions.
Icaro Lorran said: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##?