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
jth entry of
f by the
ith 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^2x*(x1)*(x13)}}
o2 =  x3+14x2+y213x 
1 1
o2 : Matrix R < R

i3 : jacobian f
o3 = {1}  3x2+28x13 
{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
