# Thread: how to prove that a parameterization parameterizes a surface....

1. ## how to prove that a parameterization parameterizes a surface....

okay this is a docarmo question asking if x(u,v) = (u+v,u-v,4uv) parameterizes the graph z=x^2 - y^2.

How do we show that it does this.

Is the question asking does the parameterization give you a regular surface that is the subset of the graph? Considering the next question says 'what parts of the surface does it cover?'

If my assumption is right.... i guess we just have to show that its 1:1 (dont need to worry about surjectivity since it only covers a part of the surface), that its cts and its inverse is cts... that its twice differentiable and that the differential is injective....

Using a theorem in docarmo, x is 1:1 implies its inverse is cts...

My big question is though:

HOW DO WE GO ABOTU SHOWING THIS PARAMETERIZES THE WHOLE SURFACE?!?

any help here would be so great!

2. ## Re: how to prove that a parameterization parameterizes a surface....

Originally Posted by mathswannabe
okay this is a docarmo question asking if x(u,v) = (u+v,u-v,4uv) parameterizes the graph z=x^2 - y^2.

How do we show that it does this.

Is the question asking does the parameterization give you a regular surface that is the subset of the graph? Considering the next question says 'what parts of the surface does it cover?'

If my assumption is right.... i guess we just have to show that its 1:1 (dont need to worry about surjectivity since it only covers a part of the surface), that its cts and its inverse is cts... that its twice differentiable and that the differential is injective....

Using a theorem in docarmo, x is 1:1 implies its inverse is cts...

My big question is though:

HOW DO WE GO ABOTU SHOWING THIS PARAMETERIZES THE WHOLE SURFACE?!?

any help here would be so great!
If you are using DeCarmo I assume you are asking whether $\bold{x}:\mathbb{R}^2\to\Gamma_f$ where $f(x,y)=x^2-y^2$ is a surface patch completely parametrizing the surface. This entails showing that $\bold{x}$ is a smooth map, and that $\text{im }\bold{x}=\Gamma_f$. Can you do thsi?