Exterior algebra of a vector bundle

Solution 1:

If $E$ is a complex vector bundle of rank $r$, its first Chern class is equal to the first chern class of its top exterior product: $$c_1(E)=c_1(\wedge ^r E)$$

This is extremely useful since the first (and only!) chern class of a line bundle is generally easy to compute.
For example on a compact Riemann surface or on a smooth projective curve, the line bundle $L=\mathcal O(D)$ associated to a divisor $D$ has its first chern class equal to the degree of the divisor: $$c_1(L)=\text {deg} D $$

Solution 2:

Tractor bundles are certain vector bundles that are more suitable than tensor bundles for the construction of invariant differential operators on some (so called parabolic) geometries.

For instance, a conformal structure $c = [g]$ on a smooth manifold $M$ defines a parabolic geometry in this sense (conformal geometry), and there exist so called (standard conformal) tractor bundle which in any choice of a metric $g \in c$ from the conformal class is just the direct sum $$ \Bbb T = \Omega^0 \oplus \Omega^1 \oplus \Omega^0 $$ but when the other metric $\hat{g} \in c$ is chosen this direct sum decomposition transforms nicely so that the (standard conformal) tractor metric and the (standard conformal) tractor connection are well-defined (invariant) on the tractor bundle.

In "Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature" of T.Branson and A.R. Gover, see e.g. here, exterior powers $$\Bbb T^k = \underbrace{\Bbb T \wedge \dots \wedge \Bbb T}_{\text{k times}}$$ of the tractor bundles were introduced (under the name of $k$-form tractors, see p. 24 there). They have a number of applications in the theory of conformally invariant differential operators.

Solution 3:

In Poisson geometry and related fields (e.g., deformation quantisation à la Kontsevich), one does actually consider the bundle $\wedge TM$ of multivector fields together with a generalisation of the Lie bracket on $\Gamma(TM)$ to $\Gamma(\wedge TM)$ called the Schouten--Nijenhuis bracket. In particular, specifying a Poisson bracket $\{\cdot,\cdot\}_M$ on a manifold $M$ is equivalent to specifying a Poisson bivector, i.e., a section $\eta \in \Gamma(\wedge^2 TM)$ such that $[\eta,\eta]=0$, via $\{f,g\}_M = (df \otimes dg)(\eta)$.