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];
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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} | a ya+zb |
{1} | 2yb xa |
{1} | 3z2c xb |
3 2
o5 : Matrix R <--- R
|