How do you express this as a product of two 1-forms?

Question

$dx\wedge dy+3\,dz\wedge dy+4\,dx\wedge dz$

I tried to do this and I reached

$dx\wedge (dy+4\,dz)+3\,dz\wedge dy$

I do not know how to proceed from here. Any hints will be appreciated.

Answer 1

You can always do this the "stupid" way which is guaranteed to work. A short calculation shows that $$ (a dx+b dy+cdz)\wedge (a'dx+b'dy+c'dz)=(ab'-a'b)dx\wedge dy+(ac'-a'c)dx\wedge dz+(bc'-b'c)dy\wedge dz$$ One solution set to these equations is $a=1,b=-\frac{3}{4},c=0, a'=0, b'=1, c'=4$. Indeed, $$ (dx-\frac{3}{4}dy)\wedge (dy+4dz)$$ works.

Answer 2

Set up the problem to find the unknown coefficients $a,b,c,p,q,r$ such that:

$$\begin{array}{rcll}(adx+bdy+cdz)\wedge(pdx+qdy+rdz)&=&&(aq-bp)(dx\wedge dy)\\&&+&(br-cq)(dy\wedge dz)\\&&+&(cp-ar)(dz\wedge dx)\\&=&&dx\wedge dy+3dz\wedge dy+4dx\wedge dz\end{array}$$

so we are trying to find $a,b,c,p,q,r$ such that:

$$\begin{array}{rcr}aq&-bp&&=&1\\&br&-cq&=&-3\\-ar&&+cp&=&-4\end{array}$$

The rest is just a routine system of linear equations, say, in unknowns $a,b,c$ with parameters $p,q,r$. Distinguish the cases and do the elimination until you find some (or all) solutions.