= Euler class of a complex vector bundle
{c}
{title2=$e(E_{\mathbb R})=c_r(E)$}
A complex rank-$r$ <vector bundle> has a canonical real orientation, and its <Euler class> equals its top <Chern class>. For a complex line this is $e(L_{\mathbb R})=c_1(L)$. The <splitting principle for complex vector bundles> gives an injective cohomology pullback on which the bundle splits into lines. The <Whitney product formula for Euler classes> and <Whitney sum formula for Chern classes> then identify both sides with the product of those first Chern classes, proving the general equality.
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