= Exterior power of a vector bundle
{title2=$\Lambda^rE$}
= Exterior powers of a vector bundle
{synonym}
= Exterior power bundle
{synonym}
For a rank-$k$ smooth <vector bundle>, take the <exterior power> $\Lambda^rE_p$ in each fibre. A local frame gives a trivialization with fibre $\Lambda^r\mathbb R^k$, and a transition matrix $A$ acts by $\Lambda^rA$. Functoriality preserves the cocycle identities, producing a smooth vector bundle of rank $\binom{k}{r}$, or the zero bundle when $r>k$. Its top exterior power is the <determinant line bundle>.
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