jacobian

jacobian R -- calculates the Jacobian matrix of the ring R
jacobian f -- calculates the Jacobian matrix of the matrix f, which will normally be a matrix with one row.
jacobian I -- compute the matrix of derivatives of the generators of I w.r.t. all of the variables

     i1 = R = ZZ/101[a..d]
     
     o1 = R
     
     o1 : PolynomialRing

     i2 = I = monomialCurve(R,{1,3,4})
     
     o2 = ideal | bc-ad c3-bd2 ac2-b2d b3-a2c |
     
     o2 : Ideal

     i3 = A = R/I
     
     o3 = A
     
     o3 : QuotientRing

     i4 = jacobian A
     
     o4 = | 0    c2   -d -2ac |
          | -d2  -2bd c  3b2  |
          | 3c2  2ac  b  -a2  |
          | -2bd -b2  -a 0    |
     
                  4       4
     o4 : Matrix A  <--- A

For a one row matrix, the derivatives w.r.t. all the variables is given

     i5 = R = ZZ/101[a..c]
     
     o5 = R
     
     o5 : PolynomialRing

     i6 = p = symmetricPower(2,vars R)
     
     o6 = | a2 ab ac b2 bc c2 |
     
                  1       6
     o6 : Matrix R  <--- R

     i7 = jacobian p
     
     o7 = | 2a b c 0  0 0  |
          | 0  a 0 2b c 0  |
          | 0  0 a 0  b 2c |
     
                  3       6
     o7 : Matrix R  <--- R

Caveat: if a matrix or ideal over a quotient polynomial ring S/J is given, then only the derivatives of the given elements are computed and NOT the derivatives of elements of J.

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