I am trying to understand how the orientation is induced on the boundary of a differentiable manifold with boundary. Here is what I have worked out so far:
Let $M$ be a differentiable manifold and let $\partial M$ denote its boundary. Let $U$ be an open set in $M$ with $U \cap \partial M \neq 0$. Similarly let $V$ be an open set in $M$ with $V \cap \partial M \neq 0$. Let $x_i$ and $y_i$ be local coordinates in $U$ and $V$ respectively. On the boundary $x_1 = 0 = y_1$. Furthermore we know that the function $y_i = y_i (x_1,\dots, x_m)$ is differentiable. Then the orientation on $\partial M$ will be the orientation given by $x_i$ and $y_i$. It remains to be shown that this is well-defined that is the same on $\partial M \cap U \cap V$. Assuming what I wrote so far is correct my question would be how to do this final step?
Note that at $x_1=0$ we have $\tfrac{\partial y_1}{\partial x_1}>0$ and $\tfrac{\partial y_1}{\partial x_i}=0$ for $i>1$. Therefore, if $x_1=0$ then the Jacobian of $(x_1,x_2,\dots,x_n)\mapsto (y_1,y_2,\dots,y_n)$ equals to $\tfrac{\partial y_1}{\partial x_1}$ times the Jacobian of $(0,x_2,\dots,x_n)\mapsto (0,y_2,\dots,y_n)$. In particular these Jacobians have the same sign. Hence everything follows.