I was working on the proof that Möbius strip is non-orientable using the following definition:
A manifold $M$ is called orientable, if there exist a smooth atlas $(U_i, \phi_i)_{i \in I}$ such that
$$\det(J(\phi_j \circ \phi_i^{-1}))> 0 \quad\text{for all}\quad i, j \in I.$$
After a couple of fails I tried to understand the proof from Riemannian Geometry by Sylvestre Gallot. He defines the following two charts on the Möbius strip $M$:
$(U_1, \phi_1)$ with $U_1 = \big( (0, 1) \times \mathbb{R} \big)/\sim$ and $\phi_1(x, y) = (x, y)$
and
$(U_2, \phi_2)$ with $U_2 = \big([0, \frac{1}{2}) \cup (\frac{1}{2}, 1] \times \mathbb{R}\big)/\sim$ and $\phi_2(x, y) = \begin{cases} (x, y), & \text{if } x > \frac{1}{2} \\ (x+1, -y), & \text{if } x < \frac{1}{2} \end{cases}$
where $(0, y) \sim (1, -y)$.
Here comes the proof from the book:
I tried to follow the proof, but I don‘t get, why should $g_j \circ \phi_1^{-1}$ (and also $g_j \circ \phi_2^{-1}$) be differentiable or at least why their partial derivatives for Jacobian exist? I don’t see any connection between $\phi_1, \phi_2$ and the atlas $(W_j, g_j)_{j \in J}$.