Orientable manifold using definition.

Question

I'm reading the Do Carmo book in the section of the orientable surfaces, but I still don't understand the idea of orientability, because the examples he uses to clarify the concept doesn't use explicitely the definition he gives, so I want to ask you if anyone could explain me with an example using the definition he uses for orientability

The definition he uses is:

A regular surface $S$ is called orientable if it is possible to cover it with a family of coordinate neighborhoods in such way that if point $p\in S$ belongs to two neighborhoods of this family, then the change of coordinates has positive Jacobian at $p$. The choice of such family is called an orientation of $S$ and $S$, in this case, is called oriented.

EDIT:

The examples Do Carmo gives are these ones:

1. A surface which is the graph of a differentiable function (cf. Sec. 2-2, Prop. I) is an orientable surface. In fact, all surfaces which can be covered by one coordinate neighborhood are trivially orientable.

2. The sphere is an orientable surface. Instead of proceeding to a direct calculation, let us resort to a general argument. The sphere can be covered by two coordinate neighborhoods, with parameters $(u, v)$ (using stereographic projection) and $(\overline{u}, \overline{v})$, in such a way that the intersection W of these neighborhoods (the sphere minus two points) is a connected set. Fix a point $p$ in $W$. If the Jacobian of the coordinate change at $p$ is negative, we interchange $u$ and $v$ in the first system, and the Jacobian becomes positive. Since the Jacobian is different from zero in $W$ and positive at $p\in W$, it follows from the connectedness of $W$ that the Jacobian is everywhere positive. There exists, therefore, a family of coordinate neighborhoods satisfying Def. I, and so the sphere is orientable.

Answer 1

The two examples shown do explicitly use the definition. But the argument is not an "equations" argument, so perhaps you won't be satisfied. I'll add some details all the same.

Example 1 is of an atlas consisting of just one neighborhood. The requirement in the definition regarding "two neighborhoods of this family" holds vacuously, because there do not exist two neighborhoods in that atlas. No equations are therefore necessary. As the author writes, the surface is "trivially orientable", because the definition holds vacuously.

Example 2 is of an atlas consisting of exactly two neighborhoods, one with parameters $(u,v)$ and the other with parameters $(\bar u,\bar v)$. The requirement in the definition "two neighborhoods of this family" therefore needs to be checked for these two neighborhoods. Now, before he checks that, there is a preliminary task to be carried out, which he describes in the sentence starting "If the Jacobian of the coordinate change at $p$ is negative... interchange $u$ and $v$...". In other words, if $$\text{det}\pmatrix{\partial u / \partial \bar u & \partial u / \partial \bar v \\ \partial v / \partial \bar u & \partial v / \partial \bar v} < 0 $$ then, swapping rows 1 and 2 with each other, we obtain $$\text{det}\pmatrix{\partial v / \partial \bar u & \partial v / \partial \bar v \\ \partial w / \partial \bar u & \partial w / \partial \bar v} > 0 $$ Thus, we now have an atlas with two coordinate charts having connected intersection $W$, and having positive Jacobian at one point $p \in W$. But the definition requires that the Jacobian be positive at all points of $W$. This follows from a topological fact: the Jacobian determinant is a nonzero continuous function on the connected space $W$ and therefore it has constant sign on $W$. Since that sign is positive at one point, namely $p$, it is positive at every point.