# Exterior derivative acting on the wedge product of three differential forms

I am familiar with the property that expands the exterior derivative acting on the wedge product of two forms,

$$d(\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^p \alpha \wedge d \beta$$

where $$\alpha$$ is a p-form and $$\beta$$ is a q-form.

Can this be generalized in the case

$$d(\alpha \wedge \beta \wedge \gamma)$$

with $$\gamma$$ being a z-form, and if yes what is the relevant formula?

And can this also be generalized for more terms in the wedge product? Something like $$d(\alpha \wedge \beta \wedge \gamma \wedge \cdots)$$?

I have tried to find the answer online but I have not been successful.

You can use the associativity of the wedge product as follows (or the other way around): $$d(\alpha \wedge \beta \wedge \gamma) = d((\alpha \wedge \beta) \wedge \gamma)$$ and then apply the graded Leibniz rule twice. (As a matter of fact, associativity is what allows you to even write such a thing as $$\alpha \wedge \beta \wedge \gamma$$.)

• oh, I see. Thanks for the clear answer!
– user494842
Mar 27, 2020 at 12:11