Different definitions for exterior derivative?

Question

Let $\omega \in \Omega^k(M)$ and $X_i \in \Gamma(TM)$. In Spivak Volume I page 213 the exterior derivative is given invariantly by the formula \begin{align*}d \omega(X_0, \ldots, X_k) = & \sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ &+ \sum_{i<j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} whereas in Kobayashi and Nomizu Volume I page 26 the exterior derivative is given by almost the same formula, but with a factor of $\frac{1}{k+1}$ in front of the sums: \begin{align*}d \omega(X_0, \ldots, X_k) = & \frac{1}{k+1}\sum_i (-1)^i X_i . (\omega(X_0, \ldots, \hat{X_i}, \ldots, X_k)) \\ & + \frac{1}{k+1} \sum_{i<j} (-1)^{i+j} \omega([X_i, X_j], X_0, \ldots, \hat{X_i}, \ldots, \hat{X_j}, \ldots, X_k) \\ \end{align*} I'm confused about where this extra factor is coming from. Even if Spivak and Kobayashi and Nomizu use different conventions for the wedge product, shouldn't the invariant formula for the wedge product be the same?

Answer 1

They both may be right. If wedge product differs (Similar problem) and we set the definition of exterior derivative as $$d\omega=\sum_{I}d\omega_I\wedge dx^I,$$ then $d$ may differs as well (cause $\wedge$ appears).

If we take axiomatic approach to exterior derivative, then one of axioms says $$d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^{|\omega|}\omega\wedge d\eta.$$ Again $d$ may differs as well (the same reason).

I just want to point out that, there is nothing strange in the fact that those two invariant formulas differs*.

*I haven't computed the invariant formulas.