Could anyone find a parameterized surface $r(u,v)$ whose normal vector at the point $(u, v)$ is
$( u - v, v - u, -2 )$?
At the moment, I've succeeded in creating $r(u, v)$ so that the partial derivative vectors are both orthogonal to the given normal vector, but their cross product is not equal to it, which also needs to happen.... So that won't work. I have also tried to create two vectors whose cross product is equal to the given normal vector and have succeeded but that hasn't worked either because I am not able to anti-derive them to create the partial derivatives of $r(u,v)$... they create some other vectors instead. So I am quite stuck. I may be thinking of this problem entirely wrong, maybe there is a pretty simple way to do this? I have not seen this type of question asked in any multivariable textbook, and I have a few. I've taught this course for 4 years and can't figure this out!! Any help would be appreciated.
Given normal vector in a plane there is no parametrized curve that can be uniquely defined. Same in 3-space.