  We argue that, under multidimensional position-dependent mass (PDM) settings,
the Euler-Lagrange textbook invariance falls short and turned out to be vividly
incomplete and/or insecure for a set of PDM-Lagrangians. We show that the
transition from Euler-Lagrange component presentation to Newtonian vector
presentation is necessary and vital to guarantee invariance. The totality of
the Newtonian vector equations of motion is shown to be more comprehensive and
instructive than the Euler-Lagrange component equations of motion (they do not
run into conflict with each other though). We have successfully used the
Newtonian invariance amendment, along with some nonlocal space-time point
transformation recipe, to extract exact solutions for a set of n-dimensional
nonlinear PDM-oscillators. They are, Mathews-Lakshmanan type-I PDM-oscillators,
power-law type-I PDM-oscillators, the Mathews-Lakshmanan type-II
PDM-oscillators, the power-law type-II PDM-oscillators, and some nonlinear
shifted Mathews-Lakshmanan type-I PDM-oscillators.
