I'm studying surface where I came along the following definition:

a coordinate patch $\displaystyle \mathbf{x}$: $\displaystyle D \mapsto \mathbb{R}^3$ is one-to-one regular mapping of an open set $\displaystyle D$ of $\displaystyle \mathbb{R}^2$ into $\displaystyle \mathbb{R}^3$

Now given this definition how would one go about checking if the given map would constitute a patch?

I'm thinking that in order to check if it's 1-1 we simply take the Jacobian and if one of the solutions is not 0 then we have a 1-1 relationship. So for instance if I have:

$\displaystyle \mathbf{x}

x,y) = (x^2,y,y^3-y)$ and I define the first set as (u,v) and the second set as (x,y,z) coordinate then the Jacobians would be:

$\displaystyle \frac{\partial(x,y)}{\partial(u.v)}$ = $\displaystyle \det \begin{pmatrix}

x_u & x_v\\

y_u & y_v

\end{pmatrix}

$ $\displaystyle = 2x \neq 0$

$\displaystyle \frac{\partial(x,z)}{\partial(u.v)}=2x*(3y^2-1)$

$\displaystyle \frac{\partial(y,z)}{\partial(u.v)}=0$

so now that at least one of the Jacobians is not 0 then it's 1-1.

is this correct?

In this particular case the last one is 0 does that make any difference?