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The answer given in my book is $x=u$, $y=v-u$, and $z=-v$. Is this essentially the same answer as mine?
There is no "absolute correct" answer, you can choose whatever you want to simplify your question.
For example, if you want to do x=u+v,y=u-v,z=-2u, it"s absolutely ok as long as it can simplify your problem.
Parametric form is just a way to simplify your integral/area.
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