# DiffOp RingElement -- apply a differential operator

## Synopsis

• Operator: SPACE
• Usage:
D f
• Inputs:
• f, , in the same ring as D
• Outputs:
• , in the same ring as f

## Description

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