DiffOp Matrix -- apply a differential operator

Synopsis

Operator: SPACE
Usage:

D f
Inputs:
- D, a differential operator,
- f, a matrix, consisting of one column, with entries in the coefficient ring of ring D, or a ring element.
Outputs:
- a ring element, in the same ring as f

Description

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 derivative 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

Ways to use this method: