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--- status: moved December 2020
-- in IntegralClosure: (jacobian, RingElement)
document {
Key => jacobian,
Headline => "the Jacobian matrix of partial derivatives",
SeeAlso => {
"diff",
"contract"
}
}
document {
Key => {(jacobian, Matrix), (jacobian, RingElement)},
Headline => "the matrix of partial derivatives of polynomials in a matrix",
Usage => "jacobian f",
Inputs => {"f" => " with one row"},
Outputs => {Matrix => {"the Jacobian matrix of partial derivatives of
the polynomial entries of ", TT "f"}},
"If ", TT "f", " is a 1 by ", TT "m", " matrix over a polynomial ring ",
TT "R", " with ", TT "n"," indeterminates,
then the resulting matrix of partial derivatives has dimensions ",TT "n"," by ",TT "m",",
and the ", TT "(i,j)", " entry is the partial derivative of the ", TT "j", "-th entry of
", TT "f", " by the ", TT "i", "-th indeterminate of the ring.",
PARA{},
"If the ring of ", TT "f", " is a quotient polynomial ring ", TT "S/J", ",
then only the derivatives of the given entries of ", TT "f", " are
computed and NOT the derivatives of elements of ", TT "J", ".",
EXAMPLE {
"R = QQ[x,y,z];",
"f = matrix{{y^2-x*(x-1)*(x-13)}}",
"jacobian f",
},
"If the ring of ", TT "f", " is a polynomial ring over a polynomial ring,
then indeterminates in the coefficient ring are treated as constants.",
EXAMPLE {
"R = ZZ[a,b,c][x,y,z]",
"jacobian matrix{{a*x+b*y^2+c*z^3, a*x*y+b*x*z}}"
}
}
document {
Key => (jacobian, Ideal),
Headline => "the Jacobian matrix of the generators of an ideal",
Usage => "jacobian I",
Inputs => {"I" => " in a polynomial ring"},
Outputs => {Matrix => {"the Jacobian matrix of partial derivatives of
the generators of ", TT "I"}},
"This is identical to ", TT "jacobian generators I", ". See ", TO (jacobian,Matrix),
" for more information.",
EXAMPLE {
"R = QQ[x,y,z];",
"I = ideal(y^2-x*(x-1)*(x-13))",
"jacobian I",
},
"If the ring of ", TT "I", " is a polynomial ring over a polynomial ring,
then indeterminates in the coefficient ring are treated as constants.",
EXAMPLE {
"R = ZZ[a,b,c][x,y,z]",
"jacobian ideal(a*y*z+b*x*z+c*x*y)"
}
}
document {
Key => (jacobian,Ring),
Headline => "the Jacobian matrix of the polynomials defining a quotient ring",
Usage => "jacobian R",
Inputs => {"R" => " a quotient of a polynomial ring"},
Outputs => {Matrix => {"the Jacobian matrix of partial derivatives of
the presentation matrix of ", TT "R"}},
"This is identical to ", TT "jacobian presentation R", ", except
that the resulting matrix is over the ring ", TT "R", ". See ", TO (jacobian,Matrix),
" for more information.",
EXAMPLE {
"R = QQ[x,y,z]/(y^2-x^3-x^7);",
"jacobian R",
},
"If the ring ", TT "R", " is a (quotient of a) polynomial ring over a polynomial ring,
then the top set of indeterminates is used, on the top set of quotients:",
EXAMPLE {
"A = ZZ[a,b,c]/(a^2+b^2+c^2);",
"R = A[x,y,z]/(a*x+b*y+c*z-1)",
"jacobian R"
}
}
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