  In this work, we study the Hermite interpolation on $n$-dimensional
non-equally spaced, rectilinear grids over a field $\Bbbk $ of characteristic
zero, given the values of the function at each point of the grid and the
partial derivatives up to a maximum degree. First, we prove the uniqueness of
the interpolating polynomial, and we further obtain a compact closed form that
uses a single summation, irrespective of the dimensionality, which is
algebraically simpler than the only alternative closed form for the
$n$-dimensional classical Hermite interpolation [1]. We provide the remainder
of the interpolation in integral form; we derive the ideal of the interpolation
and express the interpolation remainder using only polynomial divisions, in the
case of interpolating a polynomial function. Moreover, we prove the continuity
of Hermite polynomials defined on adjacent $n$-dimensional grids, thus
establishing spline behavior. Finally, we perform illustrative numerical
examples to showcase the applicability and high accuracy of the proposed
interpolant, in the simple case of few points, as well as hundreds of points on
3D-grids using a spline-like interpolation, which compares favorably to
state-of-the-art spline interpolation methods.
