I would like to determine a maximal region in $(r, \theta, z)$- space which maps injectively into $(x,y,z)$-space
Thank you
2026-04-04 23:22:04.1775344924
Maximal region in the cylindrical space
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Hint: Don't bound $z$. Bound $r$ below by $0$ (should this be strict, or non-strict?), and don't bound it above. Bound $\theta$ appropriately. (There are infinitely many ways to do this.)
Note: I am assuming, here, that "region" means "open connected set". If that isn't what you intend, let me know. I further assume that you mean for the change-of-coordinate map to be injective. If you just need a map, we could let $x=r,y=\theta,z=z$, and map the entire space injectively.