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<head><title>jacobian(Ring) -- the Jacobian matrix of the polynomials defining a quotient ring</title>
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<div><h1>jacobian(Ring) -- the Jacobian matrix of the polynomials defining a quotient ring</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>jacobian R</tt></div>
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<li><span>Function: <a href="_jacobian.html" title="the Jacobian matrix of partial derivatives">jacobian</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span>, a quotient of a polynomial ring</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the Jacobian matrix of partial derivatives of the presentation matrix of <tt>R</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>This is identical to <tt>jacobian presentation R</tt>, except that the resulting matrix is over the ring <tt>R</tt>. See <a href="_jacobian_lp__Matrix_rp.html" title="the matrix of partial derivatives of polynomials in a matrix">jacobian(Matrix)</a> for more information.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/(y^2-x^3-x^7);</pre>
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<tr><td><pre>i2 : jacobian R
o2 = {1} | -7x6-3x2 |
{1} | 2y |
{1} | 0 |
3 1
o2 : Matrix R <--- R</pre>
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If the ring <tt>R</tt> 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:<table class="examples"><tr><td><pre>i3 : A = ZZ[a,b,c]/(a^2+b^2+c^2);</pre>
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<tr><td><pre>i4 : R = A[x,y,z]/(a*x+b*y+c*z-1)
o4 = R
o4 : QuotientRing</pre>
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<tr><td><pre>i5 : jacobian R
o5 = {1, 0} | a |
{1, 0} | b |
{1, 0} | c |
3 1
o5 : Matrix R <--- R</pre>
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