### 5.1.66 jacob

`Syntax:`
  `jacob (` poly_expression `)`
  `jacob (` ideal_expression `)`
  `jacob (` module_expression `)`
`Type:`
  ideal, if the input is a polynomial
  matrix, if the input is an ideal
  module, if the input is a module
`Purpose:`
  computes the Jacobi ideal, resp. Jacobi matrix, generated by all partial derivatives of the input.
`Note:`
  In a ring with n variables, jacob of a module or an ideal (considered as matrix with a single a row) or a polynomial (considered as a matrix with a single entry) is the matrix consisting of horizontally concatenated blocks (in this order): diff(MT,var(1)), ... , diff(MT,var(n)), where MT is the transposed input argument considered as a matrix.
`Example:`

        ring R;
        poly f = x2yz + xy3z + xyz5;
        ideal i = jacob(f); i;
      ==> i[1]=yz5+y3z+2xyz
      ==> i[2]=xz5+3xy2z+x2z
      ==> i[3]=5xyz4+xy3+x2y
        matrix m = jacob(i);
        print(m);
      ==> 2yz,        z5+3y2z+2xz, 5yz4+y3+2xy,
      ==> z5+3y2z+2xz,6xyz,        5xz4+3xy2+x2,
      ==> 5yz4+y3+2xy,5xz4+3xy2+x2,20xyz3
        print(jacob(m));
      ==> 0, 2z,        2y,        2z,        6yz,5z4+3y2+2x,2y,        5z4+3y2+2x,\
         20yz3,
      ==> 2z,6yz,       5z4+3y2+2x,6yz,       6xz,6xy,       5z4+3y2+2x,6xy,       \
         20xz3,
      ==> 2y,5z4+3y2+2x,20yz3,     5z4+3y2+2x,6xy,20xz3,     20yz3,     20xz3,     \
         60xyz2

See diff; ideal; module; nvars.

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