wedge product of $0$ and $dx$

Question

I am learning differential form and I have a question about $\land$

How do we calculate $0 \land dx$? is it equal $0$? If it is, why?

Answer 1

The wedge product of a function and a one-form is just multiplication by the function. Since the function here is zero, you just get zero.

Explicitly, if $f$ is a function and $\omega$ is a one-form, then $(f\wedge \omega)_p=f(p)\omega_p.$ In this case, $f=0,$ so for any $p$, $f(p)=0.$

Answer 2

Just use that remarkable fact that $0+0=0$ and distribute: $$0 \wedge dx = (0+0)\wedge dx = 0\wedge dx + 0\wedge dx.$$Cancel $0 \wedge dx$ on both sides to get $0 = 0 \wedge dx$.