# Why does $df\wedge dg$ represent the “density of curves” of constant $f$ and $g$?

While reading the answers to Geometric understanding of differential forms., I stumbled upon this one, which, among other things, makes the case that $$df\wedge dg$$ can be interpreted as representing a "density of curves" of constant $$f$$ and $$g$$. Here $$f,g\in C^\infty(M)$$ for some manifold $$M$$.

More generally, they state that $$\alpha\wedge\beta$$, for generic 1-forms $$\alpha,\beta$$, represents the density of curves formed by the intersection of the surfaces represented by $$\alpha$$ and $$\beta$$.

I don't quite understand how this picture works. I can understand a 1-form $$\alpha$$ as characterising surfaces, in that at every point $$p\in M$$, $$\alpha_p$$ is a linear functional, and thus represents a plane via (the dual of) its normal vector. Similarly, $$df_p$$ represents a plane orthogonal to the direction of maximum variation of $$f$$, and thus I understand $$df$$ as representing surfaces of constant $$f$$.

But how does this picture work when we take exterior products?

For example, say $$M=\mathbb R^3$$ and $$\alpha=xdx+ydy$$ and $$\beta=xdx$$. Then at every point I can picture $$\alpha$$ as the plane orthogonal to the vector $$(x,y,0)$$, and $$\beta$$ as the plane orthogonal to the vector $$(x,0,0)$$. Now, $$\alpha\wedge\beta = yx \,dy\wedge dx.$$ This is now a 2-form, which takes two tangent vectors as input. How do I visualise it geometrically as a "density of curves"?

Returning to the specific case with $$df\wedge dg$$, a toy example could be $$f(x,y,z) = x+y^2, \qquad g(x,y,z) = yz, \\ df = dx + 2y \, dy, \qquad dg= z\, dy + y \, dz, \\ df\wedge dg = z \, dx\wedge dy + y \, dx\wedge dz + 2y^2 \, dy\wedge dz.$$

• This is sloppy indeed. A general $1$-form $\alpha$ in $\Bbb R^3$ does not correspond to a surface. You can get a plane at each point $p$, but very rarely will those planes integrate to give you a surface, even locally. – Ted Shifrin May 4 at 16:56
• @TedShifrin what do you mean exactly with "integrate to give a surface" here? – glS May 5 at 11:19
• For example, if $\alpha = dz - x\,dy$, there is no surface, even locally, whose tangent planes are given by $\alpha=0$. – Ted Shifrin May 5 at 16:49