Finding Curve γ(t) on M with Given Parameters

[Shackleford]

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).

Eh. I'm not quite sure how to find the curve γ(t). I think that the problem is probably a bit easier being given the parametrization of M. I do know that the point P lies in the xy-plane.

 

[BvU]

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...

 

[Shackleford]

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...

It's the tangent plane of M at P.

 

[BvU]

Good. While you are explaining anyway, is the v in x(u,v) = (u, v, u² - v²) also a point in this tangent plane ?

 

[Shackleford]

Good. While you are explaining anyway, is the v in x(u,v) = (u, v, u² - v²) also a point in this tangent plane ?

The v is in the tangent plane, not necessarily in the surface, right?

 

[BvU]

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

 

[Shackleford]

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

Yes, the v is a vector in the tangent plane, not the parameter v.

 

[BvU]

Brilliant evasion of the chore to fill in the template. As if we're among real experts.
So now you have $\gamma(0)$ as a given, yielding you u(0) and v(0).
Your turn for $\gamma'(0)$