# DiffOp -- differential operator

## Description

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