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From the book:

Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the associated ordered basis ${\{x_u,x_v}\}$.

If $p$ belongs to the coordinate neighborhood of another parametrization $\bar x (\bar u ,\bar v)$, the new basis ${\{\bar x_\bar u,\bar x_\bar v}\}$ is expressed in terms of the first one:

$\bar x_\bar u = x_u {\partial u \over \partial \bar u} + x_v {\partial v \over \partial \bar u}$

$\bar x_\bar v = x_u {\partial u \over \partial \bar v}+ x_v {\partial v \over \partial \bar v}$

(in the book the second equation is : $x_\bar v = x_u {\partial u \over \partial \bar v}+ x_v {\partial v \over \partial \bar v}$ , is a mistake, right?)

where $u=u(\bar u,\bar v)$ $v=v(\bar u,\bar v)$ are the expression of the change of coordinates.

The basis ${\{x_u,x_v}\}$ , ${\{\bar x_\bar u,\bar x_\bar v}\}$ determine, therefore, the same orientation of $T_p (S)$ if and only if the Jacobian $\partial (u,v) \over \partial(\bar u,\bar v)$ of the coordinate change is positive.

There is a proof for the last sentence?

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