Let $S$ be a surface parametrized by $\phi : \mathbb{R} ^2\to \mathbb{R} ^3$. We have that, if $\phi _u =\dfrac{\partial \phi}{\partial u}$ and $\phi _v =\dfrac{\partial \phi}{\partial v}$ then we say $\phi$ is orientation-preserving if
$\dfrac{\phi _u\times \phi _v}{||\phi _u\times \phi _v||}=n(\phi (u,v))$ for each $u,v$
where $n(\phi (u,v))$ denotes the unit normal vector to $S$ at the point $\phi (u,v)$.
Suppose $\phi :V\to\mathbb{R} ^3, \psi :W\to\mathbb{R}^3$ are two orientation-preserving parametrizations for $S$ such that there exists $G:W\to V$ differentiable and bijective and $\psi = \phi \circ G$.
I want to show that $|\det G'|=\det G'$.
I'm sure I must use that: $\det G'(\phi _u\times\phi _v) = \psi _u\times\psi _v$, but I don't know how.
Any hint? Thank you.
