I was just wondering, if we have some $k-$form $a$, and $n-$form $b$, then is it true that:
$$d(a\wedge b)=da\wedge b$$ Not; $$d(a\wedge b)=da\wedge db$$
Furthermore is it also true that if $a$ is some $k$-form defined on a manifold $M$ with dimension $2m$, then $a^k=0$ $\forall k>m?$ It says this is true, but I do not get the operation $a^k$, what does it mean for a differential form to the $k$th square mean?
Neither, if $a$ is a $k$-form and $b$ any form, then $$d(a\wedge b)=da\wedge b+(-1)^ka\wedge db.$$