Let $C$ be the cylinder $x^2+y^2=1$, let $f\colon C \rightarrow\mathbb R^3$ be the function $f(x,y,z) = (x\cos z, y\cos z,\sin z)$. Prove that the image of $f$ is precisely the unit sphere $S^2$.
Image of a function on a surface
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differential-geometry
vector-spaces