Two connex components with different orientations - The surface cannot be orientable! I need to prove that if $S $ is a surface covered by two coordinate neighborhood $V,U $ s.t. $V\cap U $ has two connex and the jacobian of coordinate change is positive in once and negative in other, so $S $ is not orientable. I cannot see how could I prove this. I appreciate some clue to beginning. Thanks in advance. ===== PS.: 1) I do not know if in English this is not used. I mean "connex components" in sense of connex pieces disconnected. 2) This is the question 1 of section 2.6 from Manfredo do Carmo's book: Differential geometry of curves and surfaces.

If $S$ was orientable, $S$ would have an atlas $\mathcal T = (T_j, \varphi_j)$ where the Jacobians of all the transition functions are positive.

Denote $C_1,C_2$ the two connected components of $U \cap V$ and take $u_i \in T_i \cap C_i$ for $i \in \\{1,2\\}$. As $S$ is path connected, consider a path $f$ joining $u_1$ to $u_2$. A chain of charts of $\mathcal T$ covering $f$ would have all positive transition functions. A contradiction with the hypothesis regarding the $U,V$ atlas.