A differential operator on the ring $R = \mathbb{K}[x_1,\dots,x_n]$ can be thought of as $k$-vectors of polynomials in $S = R[dx_1, \dotsc, dx_n]$, with coefficients in $R$, and monomials in variables $dx_1, \dots, dx_n$, where $dx_i$ corresponds to the partial derivative with respect to $x_i$. Hence a differential operator is an element of the free module $S^k$. These operators form an $R$-vector space, and operate on elements of $R^k$. The result of the operation lies in $R$, and is equal to the sum of the entrywise operations.
The ring $S$ can be obtained from $R$ using diffOpRing.
i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing |
i2 : S = diffOpRing R o2 = S o2 : PolynomialRing |
i3 : D = diffOp((x+y)*dx + (3+x) * dx*dy^2)
o3 = | (x+3)dxdy^2+(x+y)dx |
1
o3 : DiffOp in S
|
i4 : (x^2+3) * D
o4 = | (x3+3x2+3x+9)dxdy^2+(x3+x2y+3x+3y)dx |
1
o4 : DiffOp in S
|
i5 : D + D
o5 = | (2x+6)dxdy^2+(2x+2y)dx |
1
o5 : DiffOp in S
|
i6 : D(x^5*y^2)
5 2 4 3 5 4
o6 = 5x y + 5x y + 10x + 30x
o6 : R
|
i7 : D = diffOp(matrix{{x*dx}, {y*dy}})
o7 = | xdx |
| ydy |
2
o7 : DiffOp in S
|
i8 : f = matrix{{x^2}, {y^2}}
o8 = | x2 |
| y2 |
2 1
o8 : Matrix R <--- R
|
i9 : D f
2 2
o9 = 2x + 2y
o9 : R
|