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

## Synopsis

• Function: jacobian
• Usage:
jacobian f
• Inputs:
• f, , with one row
• Outputs:
• , the Jacobian matrix of partial derivatives of the polynomial entries of f

## 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