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In the parton model, the incoming parton has $k=\xi P_H$ and the outgoing massless parton has $k^{\prime }=\xi P_H+q$. Neglecting $P_H^2$ and imposing its on-shell condition givesThus the Bjorken scaling variable measures the struck parton's longitudinal momentum fraction in this approximation. The inelasticity is unchanged when the hadron momentum is replaced by its parton's collinear momentum:Let $S_H=(p+P_H{)}^2\simeq 2p\cdot P_H$. The partonic invariant in part (b) is $s(\xi )=\xi S_H$, and $Q^2=-q^2=xy{S}_H$. Insert its two parton cross sections into the parton distribution function convolution:Since the event's measured $x$ fixes $\xi =x$, the integrand is the cross-section density in $x$. No additional Jacobian from $y_q$ is needed because $y_q=y$. ThereforeEquivalently $S_H=2E_{\nu }{E}_H(1-\cos {\vartheta }_{\nu H})$ in the chosen massless-hadron frame. The explicit factor $x$ comes from the partonic energy $s=x{S}_H$, not from redefining the parton distribution functions. This expression uses the stated two-flavor and low-energy approximations, with masses and generation mixing omitted.