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Macaulay2Doc :: jacobian(Matrix)

jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix

Synopsis

Description

If f is a 1 by m matrix over a polynomial ring R with n indeterminates, then the resulting matrix of partial derivatives has dimensions n by m, and the (i,j) entry is the partial derivative of the j-th entry of f by the i-th indeterminate of the ring.

If the ring of f is a quotient polynomial ring S/J, then only the derivatives of the given entries of f are computed and NOT the derivatives of elements of J.

i1 : R = QQ[x,y,z];
i2 : f = matrix{{y^2-x*(x-1)*(x-13)}}

o2 = | -x3+14x2+y2-13x |

             1       1
o2 : Matrix R  <--- R
i3 : jacobian f

o3 = {1} | -3x2+28x-13 |
     {1} | 2y          |
     {1} | 0           |

             3       1
o3 : Matrix R  <--- R
If the ring of f is a polynomial ring over a polynomial ring, then indeterminates in the coefficient ring are treated as constants.
i4 : R = ZZ[a,b,c][x,y,z]

o4 = R

o4 : PolynomialRing
i5 : jacobian matrix{{a*x+b*y^2+c*z^3, a*x*y+b*x*z}}

o5 = {1, 0} | a    ay+bz |
     {1, 0} | 2by  ax    |
     {1, 0} | 3cz2 bx    |

             3       2
o5 : Matrix R  <--- R