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

a coordinate patch

:

is one-to-one regular mapping of an open set

of

into

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:

x,y) = (x^2,y,y^3-y)" alt="\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:

=

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?