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This is an exercise from a book called "Differential Topology"

2-11: Let $M$ be the sphere $x^2+y^2+z^2=1$ in 3-space. Prove that each of the Euclidean coordinates $x,y,z$ is a differentiable function on $M$.

I know how to show a function is differentiable, but I don't know what it means for a coordinate to be a differentiable function.

SamC
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    You need to prove that each of the natural projections $\pi_1, \pi_2, \pi_3:M \to \mathbb{R}$ is a smooth function on $M$. Where $\pi_1(x,y,z)=x$ and $\pi_2(x,y,z)=y$ and $\pi_3(x,y,z)=z$. – shalop Mar 30 '15 at 00:54
  • @Shalop So $\pi_1 (x,y,z) = \sqrt{1-y^2-z^2}$ on $M$ ? – SamC Mar 30 '15 at 02:20
  • Yeah, I guess. But I think that you just have to show that each of the $\pi_i$ is smooth, i.e, that $\pi_i \circ \varphi^{-1}$ is smooth for any coordinate chart $\varphi$ on $M$. – shalop Mar 30 '15 at 04:32

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