Moving-surface retarded Jacobian ($|1-M_r{|}^{-1}$)

For a source surface parameterized by $y(p,q,\tau )$, the retarded time equation is $t=\tau +|x-y(p,q,\tau )|/c_0$. Integrating a Dirac delta distribution enforcing that equation produces the reciprocal derivative $|1-M_r{|}^{-1}$ at each simple root. For subsonic motion there is a unique retarded root. Multiple roots require summation; a zero derivative is outside the simple-root rule. This factor is the moving-source counterpart of a delta change-of-variable Jacobian.