aka form, linear form, differential form

Exterior forms or $k$-forms are sections of an exterior power of the Cotangent bundle over manifold $X$, called the Bundle of k-forms. The set of $k$-forms is denoted $\Omega(k)$. The direct sum of $\Omega(k)$ from $k=0$ to $k=dim(X)$ is an Exterior algebra over vector bundles. The max $k=dim(X)$ is because $k$-forms for $k > dim(X)$ are $0$.

$\Omega(k) = \Gamma^\infty (\wedge^k T^* X)$

Exterior forms are Tensors.

As for exterior algebras on any vector bundles, the Wedge product is defined for exterior forms.

If $(x_1,..,x_n)$ are local coordinates on open $U \subseteq X$. Then $dx_1,...,dx_n$ (seen as vector fields on $U$) are a basis of sections of $T^* X|_U$. Hence $\{dx_{i_1}\wedge dx_{i_2} \wedge ... \wedge dx_{i_k} : 1 \leq i_1 < i_2 < ... < i_k \leq n\}$ is a basis of sections of $\bigwedge^k T^* X|_U$

One can also see differential forms as multilinear mappings (just like Cotangent vectors are linear mappings of vectors). See here and here.

Pullbacks of exterior forms

Let $f: X \to Y$ be a Smooth function of manifolds and $x \in X$ with $f(x) = y \in Y$. The Differential of $f$ at $x$ gives a linear map $T^*_x f : T^*_y Y \to T^*_x X$ between Cotangent spaces (see discussion there). As exterior products are Functorial under linear maps, this induces

$f^* := \wedge^kT^*_xf: \wedge^*_y Y \to \wedge^k T^*_x X$

This is the pullback function of exterior forms. It is Contravariantly functorial and it satisfies $f^*(\alpha \wedge \beta) = f^*(\alpha) \wedge f^*(\beta)$.

Geometric relevance

They correspond to planes, etc. See notes in Discrete geometry. Need to think more about this.

See Do Carmo – Differential forms and applications

Differential form