I want to show that on the unit sphere $S^2$, the antipodal map $A:S^2\to S^2$ given by $(x,y,z) \mapsto (-x,-y,-z)$ is orientation reversing.
I know that $S^2$ is a regular connected orientable surface so that it has two distinct orientation.
What I tried:
Consider an orientation $\{X_i(U_i)\}_{i \in I}$ where $X_i :U_i \underset { \text{open}}{\subseteq} \mathbb R^2 \to S^2$ is a familiy of parametrizations. Suppose that $AX_i(U_i) \cap X_j(U_j) \neq \emptyset$. I need to show that the determinant of the Jacobian of the change of coordinate map $X_j^{-1} AX_i$ is negative. Note $X_j^{-1} AX_i=X_j^{-1} (-X_i)$. Then, $\det \ d(X_j^{-1} AX_i)=\det\ \ dX_j^{-1}(-X_i) \circ d(-X_i)=-\det \ \ dX_j^{-1}(-X_i)\circ dX_i$. It suffices to show $\det \ \ dX_j^{-1}(-X_i)\circ dX_i >0$, but this composition is not even well defined since $dX_j^{-1}$ is evaluated at $-X_i$.
I would like to know if such an "elementary type" argument is possible. Any help is appreciated.