jacobian -- the Jacobian matrix of partial derivatives

jacobian -- the Jacobian matrix of partial derivatives

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];

i2 : I = monomialCurveIdeal(R,{1,3,4})



                        3      2     2    2    3    2

o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)



o2 : Ideal of R

i3 : A = R/I



o3 = A



o3 : QuotientRing

i4 : jacobian A



o4 = {1} | -d 0    c2   -2ac |

     {1} | c  -d2  -2bd 3b2  |

     {1} | b  3c2  2ac  -a2  |

     {1} | -a -2bd -b2  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 = {1} | 2a b c 0  0 0  |

     {1} | 0  a 0 2b c 0  |

     {1} | 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.

Ways to use jacobian :

jacobian Ideal

jacobian Matrix

jacobian MonomialIdeal

jacobian Ring