TITLE: Wedge product = set intersection?
QUESTION [8 upvotes]: In a research article [1] I found the following formulation:

The wedge product may be considered as set intersection. For
  example, surfaces of constant $f(x,y,z)$ and surface of constant $g(x,y,z)$
  intersects along the lines given by $df \wedge dg$. The notion
  of interpreting the wedge product as set intersection is appealing
  from a topological standpoint.

The article almost does not make use of exterior differential systems or other sophisticated math. It also gives no precise reference for the above statement.
Questions:

Does this viewpoint (wedge product = set intersection) make sense at all?
What is meant with "the lines given by $df \wedge dg$"? Edit: E.g. When I set $f(x,y,z) = \frac{1}{2}(x^2 + y^2 + z^2) $ and $g(x,y,z) = ax + by + cz$ with real constants $a,b,c$. Then we have $df\wedge dg = (x dx + y dy + z dz)\wedge(a dx + b dy + c dz)$. But what line does correspond to this 2-form. (I would suspect that we obtain circles.)

[1] http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=990890

REPLY [2 votes]: Define the characteristic vector (fields) of a form as $\iota_v\omega=0$. By the properties of the interior product if both $v_1$ and $v_2$ are c.v., then so are their linear combinations, $\text{span}(v_1,v_2)$. Furthermore, if the rank of the form $r$, the number of linearly independent characteristic vectors is $n-r$.
One can describe a tangent plane to $f=\text{const}$ at a regular point $p$ as a linear space $\{v\}$ such that $\iota_vdf=df(v)=f_xv^1+f_yv^2+f_zv^3=0$.
Next, using $\iota_v(\beta\wedge\gamma) = (\iota_v\beta)\wedge\gamma+(-1)^p\beta\wedge(\iota_v\gamma)$ we look for $\{v\}:\iota_v(df\wedge dg)=(\iota_vdf)\wedge dg-(\iota_vdg)\wedge df=0$. This is generally satisfied only if the vector is in the tangent spaces to both level set surfaces at points of intersection, giving us the tangent lines of the intersection. For your example, you'd have to solve simultaneously $xv^1+yv^2+zv^3=0$ and $av^1+bv^2+cv^3=0$ to get the tangent line. The intersection points, at least initial, are still to be found from $f=g=\text{const}.$