would like proof of Theorem 4-10, from Spivak's Calculus on Manifolds

Question

I'd like to understand a way to prove part 2 of this theorem.

2) If $\omega$ is a $k$-form and $\eta$ is an $l$-form, then $d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^k\omega\wedge d\eta$.

I see that this is true when $k=0$, and also when $\omega=dx^{i_1}\wedge\dots\wedge dx^{i_k}$ and $\eta=dx^{j_1}\wedge\dots\wedge dx^{j_l}$. However I haven't been able to derive the formula in general. Any help would be greatly appreciated.

Answer 1

Hint: Prove this via induction on $k + \ell$. You'll want to use the fact that any $k$-form can be written as a sum $\alpha_1\wedge \mu_1 + \cdots +\alpha_s \wedge \mu_s$, where $\alpha_1,\ldots,\alpha_s$ are $1$-forms and $\mu_1,\ldots,\mu_s$ are $(k-1)$-forms.

Answer 2

Just do it explicitly for $\omega = f dx^{i_1}\wedge\dots\wedge dx^{i_k}$ and $\eta = g dx^{j_1}\wedge\dots\wedge dx^{j_\ell}$. (It's just the usual product rule for functions.) Then the general result follows by distributing $d$ and wedge over sums.