Let $\varphi: (0, \infty) \times (0, \pi) \times (0, 2 \pi) \to \mathbb{R}^3 \setminus \{(x,y,z) \in \mathbb{R}^3| y=0, x \geq 0 \}$ $$(p,\phi,\theta) \mapsto (p \sin \phi \cos \theta, p \sin \phi \sin \theta, p \cos \phi)$$ Then $\varphi$ preserves or reverse orientation?

If $p=1$ and let $\psi: (0, \pi) \times (0, 2 \pi) \to S^2 \setminus \{(x,y,z) \in S^2| y=0, x \geq 0 \}$ $$(\phi,\theta) \mapsto (\sin \phi \cos \theta, \sin \phi \sin \theta,\cos \phi)$$ Then $\psi$ preserves or reverse orientation of sphere surface?

Let $M,N$ be manifolds and let $f:M \to N$. $f$ is called oriented preservation if for every $x \in M$, $df_x: T_xM \to T_{f(x)}N$ is a linear map preserves the orientation of Vector space.

My attempt is that I compute the determinant of the Jacobian matrix of $\varphi$ then I find it is positive, hence $\varphi$ preserves the orientation. But I don't know how to do with $\psi$? My teacher's hint is that I should use the orientation of the boundary. Can anyone help me?


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