How can I show that the parametrized torus $T=\{(x,y,z)\in \mathbb{R}^3 : (\sqrt{x^2 +y^2}-a)^2 +z^2 =b^2 \}$ is a 2-dimensional smooth submanifold of $\mathbb{R}^3$ ? I was thinking of using the regular value theorem but I can't think of a smooth map that works. Any help would be greatly appreciated.
2026-04-28 12:40:40.1777380040
Show the parametrized torus is a 2-dimensional smooth submanifold of$\mathbb{R}^3$
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Let your function be $(\sqrt {x^2+y^2}-a)^2+z^2$, with $b^2$ the regular value.