  Lebesgue integration of derivatives of strongly-oscillatory functions is a
recurring challenge in computational science and engineering. Integration by
parts is an effective remedy for huge computational costs associated with Monte
Carlo integration schemes. In case of Lebesgue integrals over a smooth
manifold, however, integration by parts gives rise to a derivative of the
density implied by charts describing the domain manifold. This paper focuses on
the computation of that derivative, which we call the density gradient
function, on general smooth manifolds. We analytically derive formulas for the
density gradient and present examples of manifolds determined by popular
differential equation-driven systems. We highlight the significance of the
density gradient by demonstrating a numerical example of Monte Carlo
integration involving oscillatory integrands.
