# preserves or reverse orientation of sphere surface

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?