User:Paul/Sandbox

Let $M$ be a smooth manifold, $T_x\left({M}\right)$ be the tangent space and $T_x^*\left({M}\right)$ be the cotangent space of $x \in M$.

A p-form on $M$ is a smooth map $\omega:M \to \bigwedge^p T^*\left({M}\right)$, where $\bigwedge$ is the wedge product and $T^*\left({M}\right)$ is the cotangent bundle of $M$. The map $\omega$ sends a point $x$ to an alternating multilinear map $\phi: \bigoplus\limits_{i \mathop = 1}^p T_x\left({M}\right) \to \R$, that takes $p$ vectors in $T_x\left({M}\right)$ and sends it to a real number.

The space of all p-forms is denoted by $\Omega^p\left({M}\right)$ and the differential operator $\mathrm{d}:\Omega^p\left({M}\right) \to \Omega^{p+1}\left({M}\right)$ is a map that sends $p$-forms to $\left({p+1}\right)$-forms.

Note that $0$-forms are simply smooth maps $\omega: M \to \R$ as a map $\phi \in \bigwedge^0 T^*\left({M}\right) = \langle{1}\rangle$ is simply a constant map.