Differentiable Manifolds/Tensor fields

Definition (Cartan derivative):

Proof: For $k\in \mathbb {N} \cup \{0\}$, it is clear that $d$ maps $\Omega ^{k}(M)$ to $\Omega ^{k+1}(M)$. We claim that also $d\circ d=0$. By linearity, we reduce to the case of a basis element, so suppose that $\omega =fdx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}\in \Omega ^{k}(M)$ with $i_{1}<\cdots <i_{k}$ and $f\in C^{\infty }(M)$. Then

${\begin{aligned}(d\circ d)(\omega )&=d\left(\sum _{j=1}^{n}{\frac {\partial f}{\partial x_{j}}}dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}\right)\\&=\sum _{k,j=1}^{n}{\frac {\partial ^{2}f}{\partial x_{j}\partial x_{k}}}dx_{k}\wedge dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}.\end{aligned}}$

By Clairaut's theorem and the anti-commutativity of $\wedge$, all terms cancel except the ones where $k=j$, and there $dx_{k}\wedge dx_{j}=0$. $\Box$