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<head><title>jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix</title>
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<div><h1>jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix</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 f</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>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, with one row</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 polynomial entries of <tt>f</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>f</tt> is a 1 by <tt>m</tt> matrix over a polynomial ring <tt>R</tt> with <tt>n</tt> indeterminates, then the resulting matrix of partial derivatives has dimensions <tt>n</tt> by <tt>m</tt>, and the <tt>(i,j)</tt> entry is the partial derivative of the <tt>j</tt>-th entry of <tt>f</tt> by the <tt>i</tt>-th indeterminate of the ring.<p/>
If the ring of <tt>f</tt> is a quotient polynomial ring <tt>S/J</tt>, then only the derivatives of the given entries of <tt>f</tt> are computed and NOT the derivatives of elements of <tt>J</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : f = matrix{{y^2-x*(x-1)*(x-13)}}
o2 = | -x3+14x2+y2-13x |
1 1
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : jacobian f
o3 = {1} | -3x2+28x-13 |
{1} | 2y |
{1} | 0 |
3 1
o3 : Matrix R <--- R</pre>
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If the ring of <tt>f</tt> is a polynomial ring over a polynomial ring, then indeterminates in the coefficient ring are treated as constants.<table class="examples"><tr><td><pre>i4 : R = ZZ[a,b,c][x,y,z]
o4 = R
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : jacobian matrix{{a*x+b*y^2+c*z^3, a*x*y+b*x*z}}
o5 = {1, 0} | a ya+zb |
{1, 0} | 2yb xa |
{1, 0} | 3z2c xb |
3 2
o5 : Matrix R <--- R</pre>
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