The discussion revolves around finding a curve γ(t) on a surface M, which is parametrized by x(u,v) = (u, v, u² - v²). The curve must pass through a specific point P=(1,1,0) on the surface and have a given tangent vector v = (7/2, 2, 3) at that point. Participants are exploring how to express the curve in terms of its parameters u(t) and v(t).
Some participants are clarifying the definitions and roles of the variables involved, particularly the tangent vector and its relationship to the surface. There is an ongoing exploration of how to derive the necessary parameters for the curve γ(t) based on the given conditions.
There is an emphasis on the need to adhere to a template for presenting the problem, which some participants note has not been fully utilized. The distinction between the tangent vector and the parameter in the surface's parametrization is under discussion.
Consider the surface M parametrized by x(u,v) = (u, v, u2 - v2) and let P=(1,1,0)\in M. Let v = (\frac{7}{2},2,3) \in T_p(M).
(a) Find a curve γ : I → M with γ(0) = P, γ'(0) = v and write γ(t) = x(u(t), v(t)), i.e. you need to find out what is u(t) and v(t).
BvU said:Et tu, Shackle: Did you notice PF has a template ?
I have no clue what you mean with ##v =(\frac{7}{2},2,3) \in T_p(M)##.
Enlighten me, and all those others who might want to help you...
BvU said:Good. While you are explaining anyway, is the v in x(u,v) = (u, v, u2 - v2) also a point in this tangent plane ?
BvU said:My guess is that there are two v floating around in the problem statement, and you need to make a distinction between them. Nice opportunity to catch up with the requirement to make use of the template!
1. The problem statement, all variables and given/known data
Homework Equations
The Attempt at a Solution