Meaning of Normal Vector in Surface Integration

Question

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!

Answer 1

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., $$|\det T|={m(TA)\over m(A)}.$$

Answer 2

Given a parametric representation $(s,t)\mapsto{\bf x}(s,t)$ of a surface $S\subset{\mathbb R}^3$ one usually reserves the letter ${\bf n}$ for the normalized surface normal vector: $${\bf n}(s,t):={{\bf x}_s\times{\bf x}_t\over|{\bf x}_s\times{\bf x}_s|}\ .$$ The vector product ${\bf x}_s\times{\bf x}_t$ itself, resp., the $2$-form $$d\omega\!\!\!\!\!\omega:={\bf x}_s\times{\bf x}_t\ {\rm d}(s,t)\ ,$$ plays a rôle in the computation of flow integrals. When the surface $S$ is embedded in a flow field ${\bf v}$ defined in ${\mathbb R}^3$ then $${\bf v}\cdot d\omega\!\!\!\!\!\omega={\bf v}\cdot ({\bf x}_s\times{\bf x}_t)\ {\rm d}(s,t)=\bigl[{\bf v},{\bf x}_s,{\bf x}_t\bigr]\ {\rm d}(s,t)$$ can be interpreted as the amount of fluid traversing the "surface element" $\>dS\>$ spanned by ${\bf x}_s\>ds$ and ${\bf x}_t\>dt$ (an infinitesimal parallelogram) per second, and the flow integral $$\int_S{\bf v}\cdot d\omega\!\!\!\!\!\omega=\int_S{\bf v}\cdot ({\bf x}_s\times{\bf x}_t)\ {\rm d}(s,t)$$ is the total amount of fluid crossing $S$ per second. Note that this integral is free of any square roots.

A parametrization $(s,t)\mapsto{\bf x}(s,t)$ with $|{\bf x}_s\times{\bf x}_s|\equiv1$ is area preserving, i.e., pieces $B$ of the $(s,t)$-parameter domain are mapped to pieces of $S$ having the same area as $B$. When $S$ is a rotational surface this can be realized in a rotationally symmetric way; whereas for an "arbitrary" $S$ nothing nice comes out of such an endeavor.