Patches and Surfaces (Differential Geometry)

[i1100]

I'm completely confused with patches, which were introduced to us very briefly (we were just given pictures in class). I am using the textbook Elementary Differential Geometry by O'Neill which I can't read for the life of me. I'm here with a simple question and a somewhat harder one.

Homework Statement

Is the following mapping x:R^2 to R^3 a patch?

x(u,v)=(u, uv, v)?

Homework Equations

For a mapping to be a patch, it must be one-to-one (injective) and regular (smooth).

The Attempt at a Solution

I understand how to show that it is regular; for any arbitrary direction, either the directional derivative of the x component or the directional derivative of the y component is non-zero. Now, I don't know how to prove that it is injective. The book gives a hint: x is one-to-one iff x(u,v) = x(u_1, v_1) implies (u,v)=(u_1,v_1).

So my attempt was to just let x(u_1,v_1) = (u_1, u_1v_1, v_1) so that
(u_1, u_1v_1, v_1)=(u, uv, v). Is this the correct way of going about it? I feel like I didn't show anything.

Can someone also point me toward a better book or online notes where I can try to understand some of this material?

Thank you, any help or suggestions will be appreciated.

 

[Tinyboss]

Suppose that x(u,v)=x(w,z). Then (u,uv,v)=(w,wz,z), and this implies (u,v)=(w,z).