Projective bundle definition of Chern classes ($h^r+{\pi }^{\ast }{c}_1(E)h^{r-1}+\cdots +{\pi }^{\ast }{c}_r(E)=0$)

For a rank-$r$ complex vector bundle, let $h=c_1(S^{\ast })$ for the tautological line $S$ on its projective bundle. The Leray-Hirsch theorem gives the free basis $1,h,\dots ,h^{r-1}$ over base cohomology. The uniquely determined coefficients in the displayed monic relation define the integral Chern classes. The line normalization uses the Euler class of the canonically oriented underlying real two-plane bundle, and uniqueness makes this definition intrinsic and natural under pullback. No choice of a trivializing cover or classifying map remains in the definition.