I would like to suggest perusing the literature of Dr. David Hestenes et al. regarding Geometric Algebra and Geometric Calculus. It soumds like you want to talk about the D' Alembertian and possibly Dirac's derivation of his equations as the inverse of 'squaring'. In what way would you interpret...
Buckminster Fuller questioned the orthogonal assumptions of multidimensional drawing and suggested instead, using 60 degree base vectors for x,y,z on one plane, with the fourth dimension as the altitude of a tetrahedron. He also preferred to refer to the second power of a number as 'triangling'...
You are not alone. See THE VECTOR CALCULUS GAP: Mathematics not= Physics
by Tevian Dray and Corinne A. Manogue (24 September 1998)
(http://www.math.oregonstate.edu/bridge/papers/calculus.pdf)
BRIDGING THE VECTOR (CALCULUS) GAP by TEVIAN DRAY and CORINNE A. MANOGUE...