Unique exterior derivative

Question

While reading that article wiki I was confused by the note saing that there is another formula for exterior derivative differing by a constant. On the other hand from what I have checked in my differential geometry book that operator is UNIQUE so how is it possible?

Answer 1

The exterior derivative is the unique operator satisfying the three axioms given in the Wikipedia article: $df$ is the differental when $f$ is a function, $d(df)=0$, and $d(\alpha\wedge\beta) = d\alpha\wedge\beta + (-1)^p\alpha\wedge d\beta$ when $\alpha$ is a $p$-form. The reason Kobayashi and Nomizu and Helgason have to use a different formula for $d\omega$ is because they use a different definition of the wedge product.