differential form

A differential k-form is a (smooth) section of the $k^{th}$ exterior power of the cotangent bundle ( i.e. ${\bigwedge }^k(T^{\ast }M)$).

In this way, for a differential $k$-form

$$\begin{array}{rl}\alpha \in {\Omega }_M^k& ⟹{\alpha }_p\in {\bigwedge }^k(T_p^{\ast }M)\\ & ⟹{\alpha }_p:((T_p^{\ast }M{)}^{\ast }{)}^k\to \mathbb{R}\\ & ⟹{\alpha }_p:(T_{p}M{)}^k\to \mathbb{R}\end{array}$$

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