If $ \eta $ and $ \varphi $ are closed differential forms, then prove that $ \varphi \wedge \eta $ is a closed differential form.

Question

Let’s assume that $ \eta $ and $ \varphi $ are closed differential forms. Then how can I prove that $ \varphi \wedge \eta $ is a closed differential form as well? Please explain how to solve this problem clearly. Thank you very much!

Answer 1

We have that $$d(\varphi \wedge \eta) = d\varphi \wedge \eta + (-1)^{\deg(\varphi)} \varphi \wedge d\eta = 0 + 0 = 0.$$ Hence $\varphi \wedge \eta$ is closed as well.