Consider the surface M parametrized by x(u,v) = (u, v, u2 - v2) and let P=(1,1,0)[itex] \in M[/itex]. Let v = ([itex]\frac{7}{2},2,3) \in T_p(M).[/itex]
(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
The purpose of finding curve γ(t) on M with given parameters is to determine a specific path on a manifold M that satisfies certain criteria or constraints.
The parameters used to define curve γ(t) on M can vary depending on the specific problem, but they typically include the starting point, ending point, and any other constraints or conditions that must be satisfied.
The curve γ(t) on a manifold M can be found using mathematical techniques such as calculus and differential equations. These methods involve minimizing a certain functional or equation in order to find the optimal path on the manifold.
Finding curve γ(t) on M is significant because it allows for the optimization of paths on a manifold, which has many real-world applications in fields such as physics, engineering, and computer graphics.
Yes, there can be limitations to finding curve γ(t) on M depending on the specific problem and the complexity of the manifold. Some problems may not have a closed-form solution and require numerical methods, and others may have multiple solutions that need to be carefully evaluated.