I was reading a textbook on differential form and it said finding the value at one point is enough to determine the orientation of the whole parameter plane at all point.
EDIT: The example they give is the sphere parameterized by
$ x=cos\theta\, cos \phi $
$ y=sin\theta\, cos\phi $
$ z=sin\phi $
for $ dy\wedge dz $ compute the pull back
$ dy= cos\theta\,cos\phi\, d\theta\,-sin\theta\,sin\phi\, d\phi$
$ dz=cos\phi \,d\phi $
$dy \wedge dz=cos\theta\,cos^2\phi \,d \theta \wedge d\phi $
and since $ x\,dy \wedge dz $ must be positive for
$[ -\frac \pi2\le \phi \le \frac \pi2 \,,\,0 \le\theta\le2\pi] $
so
$ cos^2\theta\,cos^3\phi\, d\theta \wedge d\phi\ge0 $
leave $ d\theta\ \wedge d\phi$ positive so the orientation is $ d\theta \wedge d\phi $ as orientation of $ dy\wedge dz $
but it also say that we can know the orientation by just evaluating at one point (1,0,0) and the pull back become $ 1\times cos(0)\times cos^2(0)=1 $ so $ d\theta\ \wedge d\phi$ positive oriented same as $ dy\wedge dz $.
What I don't understand is why this work and does for all parameterized surface(i.e. : a torus).
(sorry about the choice of word I'm only reading this as a hobby so don't quite really understand the term a proper mathematician might used)