Let $R = \mathbb{F}[x_1,\dots,x_n]$ and $S = R[dx_1,\dotsc,dx_n]$. The elements of $S$ operate naturally on elements of $R$. The operator $dx_i$ acts as a partial derivarive with respect to $x_i$, i.e., $dx_i \bullet f = \frac{\partial f}{\partial x_i}$, and a polynomial acts by multiplication, i.e., $x_i \bullet f = x_i f$.
Suppose $D \in S^k$ and $f \in R^k$. Then the operation of $D$ on $f$ is defined as $D\bullet f := \sum_{i=1}^k D_i \bullet f_i \in R$.
i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing |
i2 : S = diffOpRing R o2 = S o2 : PolynomialRing |
i3 : D = diffOp matrix{{x*dx}, {(y+1)*dx*dy}}
o3 = | xdx |
| (y+1)dxdy |
2
o3 : DiffOp in S
|
i4 : f = matrix{{x+y}, {x*y*(y+1)}}
o4 = | x+y |
| xy2+xy |
2 1
o4 : Matrix R <--- R
|
i5 : D f
2
o5 = 2y + x + 3y + 1
o5 : R
|
As with diffOp(Matrix), a $1\times 1$ matrix may be replaced by a ring element.
i6 : D = diffOp dx^2
o6 = | dx^2 |
1
o6 : DiffOp in S
|
i7 : D(x^3+y*x^2) o7 = 6x + 2y o7 : R |