Meaning of Normal Vector in Surface Integration Is there a good interpretation of what the normal vector (and its magnitude) $$\mathbf{N}=\frac{\partial \mathbf{X}}{\partial s}\times\frac{\partial\mathbf{X}}{\partial t}$$ to the parametric surface $\mathbf{X}(s,t)$ represents?
In line integrals the quantity $||\mathbf{x}'(t)||$ is the speed of the curve, and the way to "normalize" things is to use the arc-length parameterization.
So does there always exist (for the usual "nice" surfaces) a similar parameterization where $||\mathbf{N}||=1$? Perhaps more stringently, does there always exist a parameterization where $$\left|\left|\frac{\partial \mathbf{X}}{\partial s}\right|\right|=\left|\left|\frac{\partial \mathbf{X}}{\partial t}\right|\right|=1\;\;\text{and}\;\;\frac{\partial \mathbf{X}}{\partial s}\cdot \frac{\partial \mathbf{X}}{\partial t}=0$$ and if so, what would this mean? Is this some kind of "orthogonal unit speed" parameterization or something? Cheers!

The direction is obvious ($\bf N$ is orthogonal to the tangent plane), but the magnitude is definitely nontrivial: is the local factor of measure change of the parametrization. Is like $\|x'(t)\|$ in curves and abs(jacobian) in the change of variables formula.
EDIT:
The easier case in the change of variables formula is: $A\subset\Bbb R^n$ open, $T:\Bbb R^n\longrightarrow\Bbb R^n$ linear, $m=$Lebesgue measure.
$$|\det T|\,m(A)=\int_A|\det T|=\int_{TA}1=m(TA),$$
i.e.,
$$|\det T|={m(TA)\over m(A)}.$$
