Suppose I want to find a 2-form in ${\bf R}^3$ which gives the surface area to the paraboloid $D$ in some bounded region,
$$\phi(s,t)=(s,t,s^2+t^2)$$
I know that a general 2-form in ${\bf R}^3$ has the form,
$$\omega=f_1dx\wedge dy + f_2dy\wedge dz+f_3dx\wedge dz$$
Then this becomes,
$$\int_D\omega=\int_{\phi^{-1}(D)}\Big[f_1(\phi(s,t))-f_2(\phi(s,t))2s+f_3(\phi(s,t))2t\Big]ds\wedge dt$$
$$$$
If $J$ is the Jacobian for $\phi$, then since I know that the surface area element is given by $\sqrt{\det(J^TJ)}=\sqrt{1+4s^2+4t^2}$, then that means one possible solution would be,
$$f_1=f_2=f_3=\frac{\sqrt{1+4x^2+4y^2}}{1-2x+2y}$$
More generally, for surfaces in ${\bf R}^3$, it seems like one can always determine a (the?) surface area 2-form by computing,
$$f_1=f_2=f_3=\frac{\sqrt{(\det J_{xy})^2 + (\det J_{yz})^2 + (\det J_{xz})^2}}{\det J_{xy} + \det J_{yz} + \det J_{xz}}\circ\phi^{-1}$$
Clearly any time a parametrization can be formed by simply projecting down onto the xy-plane, computing this is trivial, however I imagine that computing $\phi^{-1}$ in general may be intractable. Nevertheless, $\phi$ is injective, so should it not always exist in theory?
Moreover, what exactly is this object? When can it be computed? does it always in theory exist? And is this generalizable to any dimension.
Best Answer
In general, the surface area $2$-form will be $$\omega = n_1\,dy\wedge dz + n_2\,dz\wedge dx + n_3\,dx\wedge dy,$$ where $\vec n$ is the unit outward-pointing normal. I've derived this in other posts on here.