The differential operators of the ring $R = \mathbb{F}[x_1,\dots,x_n]$ act naturally on elements of $R$. The operator $dx_i$ acts as a partial derivarive with respect to $x_i$, and a polynomial acts by multiplication.
i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing |
i2 : dx = diffOp{x^2 => 1}
2
o2 = dx
o2 : DiffOp
|
i3 : D = diffOp{1_R => x^2 + y^2}
2 2
o3 = x + y
o3 : DiffOp
|
i4 : dx(x^4 + x^3 + y)
2
o4 = 12x + 6x
o4 : R
|
i5 : D(x^2 - y^2)
4 4
o5 = x - y
o5 : R
|